# Telescoping Series Partial Fractions

Telescoping Series and Partial Fractions. Use n = 3, since we're after the 3rd partial sum. You may want to review that material before trying these problems. 003+cdots#. The first part is 2/1, so thats 2. Ask a question. Geometric and telescoping series The geometric series is X1 n=0 a nr n = a + ar + ar2 + ar3 + = a 1 r provided jrj<1 (when jrj 1 the series diverges). be able to recognize a telescoping series, determine whether it is cvg or div and if cvg, to what. Determine convergence, absolute convergence and divergence of infinite series using the standard convergence tests. The partial fraction decomposition will consist of one term for the factor and three terms for the factor. You don't see many telescoping series, but the telescoping series rule is a good one to keep in your bag of tricks — you never know when it might come in handy. Now, we need to be careful here. Could anyone explain in plain english whats going on here? And clarify their formula. In this case, P a n = s. BC Sequences and series 2015. 4 Comparison Tests. Telescoping series. • Define what a power series is and be able to find the radius and interval of convergence for a given series. Order of Operations Factors & Primes Fractions Long Arithmetic Decimals Exponents & Radicals Ratios & Proportions Percent Modulo Mean, Median & Mode Algebra Equations Inequalities System of Equations System of Inequalities Basic Operations Algebraic Properties Partial Fractions Polynomials Rational Expressions Sequences Power Sums Induction. a) Using partial fractions, , where A — n2—1 n —1 n +1. (a) Let sN be the Nth partial sum, sN = PN n=2 1 n2−n. It is called a telescoping series when you write out the sequence of partial sums and the middle terms all cancel,1381. (MCMC 2009I#4) Find the value of the in nite product 7 9 26 28 63 65 = lim n!1 Yn k=2 k3 1 k3 + 1 : Solution. Find the limit of a given sequence. We recover Theorem 1 from Theorem 6 as the case a1 = 1, ai = bj = 0 for all i > 1,j ≥ 1. Say we have something we want to sum up, let's call it a k. partial fractions to rewrite it as a telescoping series. So capital N, so the partial sum is going to be the sum from n equals one but not infinity but to capital N of negative 1 to the n minus 1. 10-2-43 telescoping series. Can the series be compared favorably to one of the special types?. 1A1: The nth partial sum is defined as the sum of the first n terms of a sequence. Laval Kennesaw State University Abstract This hand out is a description of the technique known as telescoping sums, which is used when studying the convergence of some series. Indicate a good method for evaluating the integral. Contractive sequences converge. Make sure you can correctly answer questions involving telescoping series and partial sums. There is no test that will tell us that we've got a telescoping series right off the bat. Write out the first three partial sums of 2 3 4 n n 4 b. Thus a telescoping series will only converge if bn approaches a finite limit. Series and Partial Sums. We explain calculus and give you hundreds of practice problems, all with complete, worked out, step-by-step solutions. Here, (using partial fractions) Partial sum: Thus the series converges and has sum = 5. Partial sums Value of infinite series as limit of sequence of partial sums Convergence vs. More examples can be found on the Telescoping Series Examples 1 page. Topic 9: Calculus Option Series Part 1 A series consists of C v+C w+C x…qK S=z T St_ qK s=z S T st_ It is denoted by: For a total sum For a partial sum The Divergence Test states:. Telescoping Series. can the integral test, the root test, or the ratio test be applied? 4. 10-3-33 sine series that diverges by nth term test. For example, is a partial fractions decomposition of. Telescoping Series - Sum. Improper integrals of type 2. Try this telescoping sum. Math 222, Calculus and Analytic Geometry 2, 2013-14 Notes for how to do partial fractions updated; This is some notes about infinite series and telescoping. 4 – Divergence & Integrals -Use the divergence test to determine whether a given series diverges (or whether the test is inconclusive). ALTERNATING SERIES Does an =(−1)nbn or an =(−1)n−1bn, bn ≥ 0? NO Is bn+1 ≤ bn & lim n→∞ YES n =0? YES an Converges TELESCOPING SERIES Dosubsequent termscancel out previousterms in the sum? May have to use partial fractions, properties of logarithms, etc. 1, etc), then:. The Partial Fraction Decomposition Calculator an online tool which shows Partial Fraction Decomposition for the given input. diverges ￻ ￹ c. Once it has been determined that a given series converges, it is usually a much more diﬃcult problem to actually determine the sum of the series. Partial Fractions Telescoping Series, and Harmonic Series in this section. It is called a telescoping series when you write out the sequence of partial sums and the middle terms all cancel,1381. We know how to sum all these geometric series, so we get. The comparison tests are used to determine convergence or divergence of series with positive terms. And if the limit of the partial sum is nite, then it converges, and we. In the cases where series cannot be reduced to a closed form expression an approximate answer could be obtained using definite integral calculator. To be able to do this, we will use the method of partial fractions to decompose the fraction that is common in some telescoping series. Find the power series of functions, determine their radius and interval of convergence,. It is capable of computing sums over finite, infinite (inf) and parametrized sequencies (n). (1) and ask whether the sum is convergent. First, note that the telescoping series method only works on certain fractions. — Try Before you Buy To start a Maplet, click on its name. Here’s an example series: Just like for sequences, we want to make explicit what the term is. If you update to the most recent version of this activity, then your current progress on this activity will be erased. In general, the series given by. Question is : Determine whether the series is convergent or divergent by expressing Sn as a telescoping sum. This calculator will find the sum of arithmetic, geometric, power, infinite, and binomial series, as well as the partial sum. Telescoping Series This next series is a clever series, called a telescoping series. Find the power series of functions, determine their radius and interval of convergence,. Find the sum of the series • Â k=0 4 k2 +3k+2 if it exists. 12 is an example of a telescoping series. Download this MAT 1348 class note to get exam ready in less time! Class note uploaded on Oct 14, 2019. 3 Telescoping Series. Even after 1,000,000 1,000,000 terms, the partial sum is still relatively small. Chapter 2 Infinite Series Page 1 of 7 Chapter 2 : Infinite Series Section B Telescoping and Harmonic Series By the end of this section you will be able to • use partial fractions to test for convergence and determine the sum of a series. We try This is an example of a telescoping series. Example diverve Determine if the following series serjes or diverges. For example, 1 + 1/2 + 1/3 is a partial sum of the first three terms. Welcome to Maplets for Calculus. 2) Expand first 5 terms. partial fractions to rewrite it as a telescoping series. Maclaurin and Taylor Series g. A telescoping series. Don't quite understand how to do that. 1 An introduction to series and sequences 245 261; 10. -To determine whether or not this series converges and to find what value it converges to, we need to use the telescoping series. Multiplying by a Constant Property. In mathematics, a telescoping series is an informal expression referring to a series whose sum can be found by exploiting the circumstance that nearly every term cancels with either a succeeding or preceding term. be able to recognize a telescoping series, determine whether it is cvg or div and if cvg, to what. The second part as it goes to infinity is 0, so the answer would be 2? I'm really confused and looking up examples of telescoping series and it is not helping. ) Telescoping series: A telescoping series is a series of the form P∞ n=1 (a n+i −a ) for some integer i ≥ 1. Necessary conditions for convergence. Geometric series: Converges if jrj< 1, diverges if jrj 1. Convergence Tests d. One of these tests is the telescoping series test. In this video from PatricJMT we look at why when evaluating a telescoping series, one typically finds an expression for a partial sum, and then takes the limit of this partial sum. A telescoping series (or telescoping sum) is one that "expands" in such a way that most of its terms cancel away. Use the Integral Test to decide whether the in nite series X1 n=1 2n n2 + 1 dx. Regardless, your record of completion wil. 4 e) Use partial fraction decomposition to decompose n n nn S S S S nn f f. In mathematics, a telescoping series is an informal expression referring to a series whose sum can be found by exploiting the circumstance that nearly every term cancels with either a succeeding or preceding term. [Hint: Telescoping - Use partial fraction decomposition] Solution We recognize the series as a telescoping series, so we want to construct a formula for the Nth partial sum, S N, and nd its limit as N !1. I know the answer is 3/4 but I can't figure out how to get to that answer. Get the free "Series Calculator" widget for your website, blog, Wordpress, Blogger, or iGoogle. 1674 terms are needed for the partial sum to exceed 8. p-Series 4. Telescoping series. For example, 1 + 1/2 + 1/3 is a partial sum of the first three terms. The cancellation technique, with part of each term cancelling with part of the next term, is known as the method of differences. Calculus 2 Syllabus. Hence, Therefore, by the definition of convergence for infinite series, the above telescopic series converges and is equal to 1. More examples can be found on the Telescoping Series Examples 2 page. Then a partial fraction decomposition of is so that (This summation is a telescoping sum. The first part is 2/1, so thats 2. Improper integrals of type 2. Differentiating and Integrating Power Series. Also see pages 5 through 9 of the Class Notes Day 26 on Partial Fractions. 4 Comparison Tests: Test #1-in class. Telescoping: Transform by partial fractions Procedure for Determining Convergence No Series Diverges nth-Term Test Yes or maybe Yes No Yes Nonnegative terms and/or absolute convergence No Yes No Can the Integral Test, the Ratio Test, or the Root Test be applied? Does the Integral Test apply? Yes Does the Ratio Test apply? No Yes Is Is Does the. • Determine if a series is absolutely convergent. and B b) Determine whether the following telescoping series is con- vergent or divergent. The series in Example 8. I used partial fractions to get ((1/3)/n - (1/3)/(n(n+3))). Download this MAT 1348 study guide to get exam ready in less time! Study guide uploaded on Oct 14, 2019. In many cases it is possible at least to determine whether or not the series converges, and so we will spend most of our time on this problem. Find a formula for the nth partial sum s nof the series, and use the formula for s nto determine whether the series converges. Worksheet 2 – Special Series (Telescoping) Name _____ Find the partial fraction decomposition of each of the following. Ask a question. Scope and Sequence Unit 1 - Techniques of Integration This unit includes the chain rule, u-substitution, expanding, separating the numerator, completing the square, dividing, adding and subtracting terms, trig identities, integration by parts, trig integrals, trig substitution, partial fractions, L'Hopital's Rule, improper integrals, and Euler's Method. can the integral test, the root test, or the ratio test be applied? 4. In this case it’s , so we can write the series as (or just if we’re not too concerned with where the series starts). Telescoping series are not very common in math-ematics but are interesting to study. From Wikipedia, the free encyclopedia. Math 296: Calculus II. State the de nition of geometric series. FIND THE SUM OF TELESCOPING SERIES? Use partial fraction decomposition to write 1/ you can replace the single fraction with the sum of fractions and you have. The Partial Fraction Decomposition Calculator an online tool which shows Partial Fraction Decomposition for the given input. For n = 1, the series is a harmonic series 1 2 + 1 3 + 1 4 + 1 5 + which is divergent, and the formula 1=(n 1) would indicate that the series should be divergent. Z 8x p (telescoping series) 12. 3 cont'd Integral test for series with non-negative terms cont'd and estimations, Estimations of the value of the series. f ¦ diverges. The number and variety of exercises where the student must determine the appropriate series test necessary to determine convergence of a given series has been increased. Partial fractions - Integrating rational functions is an extremely entertaining activity. Before looking for the partial fraction decomposition of the ra-. Series Geometric Senes Telescoping Series Limaçons and Lemniscates Parametric Equations—Second Derivatives and Tangent Lines Partial Fractions Convergence and Divergence Series Indexing Arithmetic of Series Integration by Parts Il Vector Functions Implicit Differentiation Il Infinite Limits of Integration Partial Fractions Ill p-Series. Infinite Series, Geometric Series, Telescoping Series; Integral Test, p-series, and Estimates of Sums; The Comparison Tests; Alternating Series and Estimates of Sums; Absolute and Conditional Convergence, Ratio and Root Tests; Strategy for Testing Series, Summary of Convergence Tests; Power Series; Representation of Functions as a Power Series. EXAMPLE 8:. The simplest way would be to use partial fractions, and then convert this into a telescoping series. These types of series frequently arise from partial fraction decompositions and lead to very convenient and direct summation formulas. Begin by using partial fraction decomposition to obtain 2 n(n+ 2) = A n + B n+ 2 = 1 n 1. 10-2-43 telescoping series. The following series, for example, is not a telescoping series despite the fact that we can partial fraction the series. Telescoping series: Compute the nth partial sum, sn, and take the limit of sn as n goes to 1. The series in Example I(b) is a telescoping series of the form — 192) + (b2 — 173) + (b 3 — 125) + Telescoping series Note that b2 is canceled by the second term, 173 is canceled by the third term, and so on. Assignment for Day 26 and Answers. Geometric Series + 107. Here's another example that uses partial fractions. High School AP Calculus BC Curriculum. Example 4: Determine the series 1 1 n nn( 1) f ¦ is convergent or divergent. Theorem 4. A more general technique Efthirniou's technique can be generalized to series of the form CrI1unvnwhere it is convenient to write only vnas a Laplace transform integral. 12 is an example of a telescoping series. (Telescoping series) Find. if it is convergent, find its sum. First, note that the telescoping series method only works on certain fractions. Partial Fractions. This free online course offers you some Integral definitions, Explanations and Examples. If Partial fractions gives 1 k(k +1). Telescoping series. o q bASl BlB Zr niVg8hnt osS 5r8ewsXenrZv Yecdj. 2 Series A series is a sum of sequential terms. 2) Trigonometric Substitution (7. Harmonic series. Next, prove an important convergence theorem. com allows you to find the sum of a series online. 10-3-23 constant times a divergent series. Expanding the sum yields Rearranging the brackets, we see that the terms in the infinite sum cancel in pairs, leaving only the first and lasts terms. Download this MAT 1348 class note to get exam ready in less time! Class note uploaded on Oct 14, 2019. The value of the series is lim I lim Sn 553 Related 47-58 19—34. Then evaluate lim n!1 S n to obtain the value of the series, or state that it diverges. MCrawford (20:20:34). The second part as it goes to infinity is 0, so the answer would be 2? I'm really confused and looking up examples of telescoping series and it is not helping. Series Calculator computes sum of a series over the given interval. Substitution. Convergence b. How this Calculus 2 course is set up to make complicated math easy: This approximately 200-lesson course includes video and text explanations of everything from Calculus 2, and it includes more than 275 quiz questions (with solutions!) to help you test your understanding along the way. Too bad the book didn't do a good job on teaching how to find nth partial sum formula. That is, s k → -∞. Geometric and Telescoping Series In this worksheet, you will calculate the exact values of several series. The voice explains how to first plug in the numbers given for each variable in the fractions. Telescoping Series 1 1 2 3 3 Show that the telescoping series ln dive rges. Telescoping series is a series where all terms cancel out except for the first and last one. Absolute convergence. A telescoping series is a special type of series whose terms cancel each out in such a way that it is relatively easy to determine the exact value of its partial sums. In algebra, the partial fraction decomposition or partial fraction expansion of a rational function (that is, a fraction such that the numerator and the denominator are both polynomials) is an operation that consists of expressing the fraction as a sum of a polynomial (possibly zero) and one or several fractions with a simpler denominator. Exercise 6. The method of partial fractions may reveal these. Your explanation that the series is telescoping after written as a partial fraction is more concise. An in nite series can be represented as such: P 1 n=1 a n. z T BMAapdPeB wwMitEhL lIQnkfoimnBi\tieE rPCrvecWavlccfuxlluKsx. Telescoping Series Test Series: ∑∞𝑛=1 Ὄ 𝑛+1− 𝑛Ὅ Condition of Convergence: lim 𝑛→∞ 𝑛=𝐿 Condition of Divergence: None NOTE: 1) May need to reformat with partial fraction expansion or log rules. The typical example of telescoping series (for partial fractions) is. There is no way to actually identify the series as a telescoping series at this point. Math - Calculus II SERIES ( Partial Sums ) infinite series GEOMETRIC SERIES Otherwise, the series diverges ( that is, the series has no sum ). Let’s decompose it, using partial fractions: [math]\frac{1. The Telescoping Series! This type of infinite series utilizes the technique of Partial Fractions which is a way for us to express a rational function (algebraic fraction) as a sum of simpler fractions. Could anyone explain in plain english whats going on here? And clarify their formula. This is the main origin of the name telescoping series. If convergent find the sum, and if divergent enter DIV: 11. This is an excerpt from my full length lesson. The final answer is: Note that we have converted an infinite sum problem to adding up a finite number of fractions. be able to recognize a telescoping series, determine whether it is cvg or div and if cvg, to what. You don't see many telescoping series, but the telescoping series rule is a good one to keep in your bag of tricks — you never know when it might come in handy. \ B jArlnlA Er^iOgqhEtcsn srhemsNeKrkvre_dM. partial fractions to rewrite it as a telescoping series. Here’s an example series: Just like for sequences, we want to make explicit what the term is. The final answer is: Note that we have converted an infinite sum problem to adding up a finite number of fractions. After several hours of thinking about these type of problems I found out that I can separate the 1/[(n+1)(n+2)] into partial fractions, and I found out that it is a telescoping series. Telescoping Series with Partial Fractions Mathispower4u. The solution in the above example uses the method of partial fractions to try to calculate the sum of a telescoping series. Then evaluate lim n!1 S n to obtain the value of the series, or state that it diverges. The following exercises test your understanding of infinite sequences and series. • Use improper integrals to reinforce the notions of infinite series.    A telescoping series is one whose partial sums eventually only have a fixed number of terms after cancellation. The result is a simple formula for the nth term of the sequence of partial sums. Her practical approach makes math manageable for anyone who's learning math for the first time, returning to school after a long break, or for anyone with a genuine curiosity who wants to dive deeper into the material. 1B3: If a series converges absolutely, then any series obtained from it by regrouping or rearranging the terms has the same value. Now we must solve for A and B. In particular, in order for the fractions to cancel out, we need the numerators to be the same. (Of course, an infinite geometric series is a special case of a Taylor series. Note: For an example of a telescoping sums question, see question #2 in the Additional Examples section below. Berndt also states that there is no “natural” way to obtain an expansion of H n in powers of m. Telescoping series: Expand out the sums to find a grouping pattern that can simplify in order to take the limit of S n *Note: Some series can be turned into subtraction via partial fractions or by log rules ∑Eg: ∑. Infinite Series Definition. This is an example of a telescoping series. Partial Fractions Introduction to Partial Fractions Linear Factors Irreducible Quadratic Factors Improper Rational Functions and Long Division Summary Strategies of Integration Substitution Integration by Parts Trig Integrals Trig Substitutions Partial Fractions Improper Integrals Type 1 - Improper Integrals with Infinite Intervals of Integration. If instructions say to determine if a series converges or diverges AND find the sum if it converges, then it must be an infinite geometric series or a telescoping. Improper integrals of type 2. If convergent find the sum, and if divergent enter DIV: 11. Geometric and Telescoping Series c. Look at the partial sums: because of cancellation of adjacent terms. So capital N, so the partial sum is going to be the sum from n equals one but not infinity but to capital N of negative 1 to the n minus 1. Geometric Series + 107. If an input is given then it can easily show the result for the given number. How to get the partial fractions of higher degree numerators. An in nite series can be represented as such: P 1 n=1 a n. The Sigma notation for summation of series. 3 The Integral Test: The section includes the integral test and a discussion on the remainder estimate for the integral test. Review for Test 3 Math 1552, Integral Calculus The series is telescoping, so it converges and we can ﬁnd its sum using partial fractions: ". Say we have something we want to sum up, let's call it a k. Be sure to review the Telescoping Series page before continuing forward. Partial Fractions. For the telescoping series X1 k=3 2 (2k 1)(2k + 1); nd a formula for the nth term of the sequence of partial sums fS ng. I work through an example of proving that a series converges and finding the sum of the series using Partial Fractions to create a Telescoping Series. (1) and ask whether the sum is convergent. ~~~~~ AP Cal BC Students: Telescoping Series and Partial Fractions Tests. A telescoping series (or telescoping sum) is one that "expands" in such a way that most of its terms cancel away. ) (Now evaluate the limit. 2: p 2] EXAMPLES Does the series converge or diverge? 1. Definitions: Let{ } 1 n n a. Retrieved from " https: The following series, for example, is not a telescoping series despite the fact that we can partial fraction the series terms. Alternating Series Test cannot be applied ￻ 26. In nite Series SUGGESTED REFERENCE MATERIAL: As you work through the problems listed below, you should reference your lecture notes and the relevant chapters in a textbook/online resource. TELESCOPING SERIES Now let us investigate the telescoping series. ) = 1 - 0 = 1. So: s1=a1 s2=a1+a2 s3=a1+a2+a3 s4=a1+a2+a3+a4. This is a challenging sub-section of algebra that requires the solver to look for patterns in a series of fractions and use lots of logical thinking. Improper integrals of type 2. This time there will be a few more terms that do not cancel. Lecture 17: Series (II) Telescoping Series A telescoping series is a special type of series for which many terms cancel in the nth partial sums. (The first Maplet may take a little longer to open because it needs to start Java. Back to Thomas O'Sullivan's Homepage. Creating the telescoping effect frequently involves a partial fraction decomposition. Telescoping series. 3 Telescoping Series. You - ProProfs Discuss. We will now look at some more examples of evaluating telescoping series. If you are looking for more in partial fractions, do check in: Partial fractions of lower degree numerators. Get the free "Series Calculator" widget for your website, blog, Wordpress, Blogger, or iGoogle. Partial fractions Fractions in which the denominator has a quadratic term Sometimes we come across fractions in which the denominator has a quadratic term which. Be able to determine the nth partial sum of any geometric series. Partial Fractions Stewart Chapter 7. Test for convergence of a series using an appropriate test: divergence, integral, comparison and limit comparison, ratio, root, or alternating series. High School AP Calculus BC Curriculum. For telescoping series, intermediate terms 'drop. For example, using partial fractions and cancelling a bunch of terms, we find that; An infinite series that arises from Parseval’s theorem in Fourier analysis. Most telescopic series problems involve using the partial fraction decomposition before expanding it and seeing terms cancel out, so make sure you know that very well before tackling these questions. Calculus 2 Syllabus. Note that the fundamental concepts of functions, graphs, and limits, which are studied at the beginning of courses in differential calculus, are often first introduced in earlier classes (most notably intermediate algebra and precalculus). Also note that it is possible to tell that this last series. Inﬁnite series 1: Geometric and telescoping series Main ideas. The final answer is: Note that we have converted an infinite sum problem to adding up a finite number of fractions. 26, 29, 39, 40 21 – 42 Solve applications involving series. Exercise 6. 3) Cancel duplicates. If you do not own Maple, click Use MapleNET 12 at the top-right corner of this page. Note: is a telescoping series. Thus, as $$k$$ increases, the partial sum $$S_k$$ increases (the series is a sum of positive terms), but is always smaller than \(2\text{. Find the sum of the series • Â k=0 4 k2 +3k+2 if it exists. That is, s k → -∞. Hence, Therefore, by the definition of convergence for infinite series, the above telescopic series converges and is equal to 1. Find the sum of the series • Â k=1 1 k 1 k+2 if it exists. With geometric series, we carried out the entire evaluation process by finding a formu for the sequence of partial sums and evaluating the limit of the sequence. Infinite Series Chapter 1: Sequences and series Section 4: Telescoping series Page 2 We can use the method of partial fractions to rewrite the general term as: 4 1 1 2 2 2 2 ab n n n n n n §· ¨¸ ©¹ For your practice, check that this decomposition is correct! Now we can write any partial sum of this series as: 11 4 1 1 2 22 kk nn n n n n. This video is a great one on learning about evaluating fractions. Telescoping: Transform by partial fractions Procedure for Determining Convergence No Series Diverges nth-Term Test Yes or maybe Yes No Yes Nonnegative terms and/or absolute convergence No Yes No Can the Integral Test, the Ratio Test, or the Root Test be applied? Does the Integral Test apply? Yes Does the Ratio Test apply? No Yes Is Is Does the. (a) X1 n=3 1 2n 1 1 2n 4 As I stated in my email follow-up to this problem, this is not actually an easy series to nd a. Now pop in the first term (a 1) and the common ratio (r). The telescoping Series is a method for examining the convervence of infinite series of the form: This method, combined with partial fraction decomposition, is frequently effective. Be able to determine the nth partial sum of any geometric series. (a) X1 n=1 cos 1 n cos 1 n+ 1. Infinite series allow us to add up infinitely many terms, so it is suitable for representing something that keeps on going forever; for example, a geometric series can be used to find a fraction equivalent to any given repeating decimal such as: #3. Besides finding the sum of a number sequence online, server finds the partial sum of a series online. According to Ferraro [4], Mengoli did so by making frequent use of the following. — Try Before you Buy To start a Maplet, click on its name. if integral of series reaches ∞ or DNE then series diverges. Telescoping Series and Partial Fractions. In this case, the series is convergent and the sum isS lim n→ sn b1 −lim n→ bn 1. See if you can ﬁgure it out. Welcome to Maplets for Calculus. 4 Comparison Tests. Can the series be compared favorably to one of the special types?. EXAMPLE 8:. 50 46 – 53 Evaluate telescoping series. telescoping series to ﬁnd out. This is best demonstrated with an example. I work through an example of proving that a series converges and finding the sum of the series using Partial Fractions to create a Telescoping Series. Now let us investigate the telescoping series. (a)Sequences and Series, partial sums. Example: Determine whether the given series converge. Use comparison test if numerator fluctuates between two constants. From Wikipedia, the free encyclopedia.