Roberto s Notes on Infinite Series Chapter 1: Sequences and series Section 4. Telescoping series. Clear as mud!

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1 Roberto s Notes on Infinite Series Chapter : Sequences and series Section Telescoping series What you need to now already: The definition and basic properties of series. How to decompose a rational expression into partial fractions. What you can learn here: How to determine the convergence of special series whose terms can be split into differences. In the last section we looed at a type of series that is very common, very useful and whose convergence properties are totally nown. In this section you will loo at another type of series that is not very common, actually rather special, is only useful in a limited number of situations and whose convergence depends in a delicate way on the individual series. That s quite a change! Yes, but we shall also use a classic and familiar method of algebra. Here is the general idea. different values of n in the partial sums to produce a simple expression for S. The limit of the simpler expression for easy to compute. S, so as S is In such a case the series is said to be a telescoping series. Definition Consider a series of the form n t n, where: each function tn can be written as a difference of similar looing terms some of these terms cancel each other for Clear as mud! As I said, this is a method that only wors in special cases, so the two examples I am about to offer will probably shed more light than the formal definition. n Example: n n The terms of this series come from a rational function of n, so we try to loo at their partial fractions decomposition. Infinite Series Chapter : Sequences and series Section : Telescoping series Page

2 We can use the method of partial fractions to rewrite the general term as: a b n n n n n n For your practice, chec that this decomposition is correct! Now we can write any partial sum of this series as: n n n n n n Notice that all fractions cancel in pairs, except for four of them: As increases to infinity this quantity becomes: S lim 3 And therefore the series converges to this value sin sin sin sin sin sin sin sin S sin sin sin sin sin sin As you can see, there are pairs of terms that are opposite and cancel. The only ones we are left with are: Therefore: S sin sin sin sin sin sin n n n lim sin sin sin sin n n n sin sin sin 0 sin 0 sin sin In conclusion, the series converges to this value. The method can also be applied for other series in which the cancellations come from the very definition of the terms. Example: n sin sin n n We can write a partial sum of this series as: Infinite Series Chapter : Sequences and series Section : Telescoping series Page

3 Summary Some special series can be rewritten so that their partial sums simplify to expressions whose limit at infinity can be easily computed. These series are called telescoping and their convergence and limit may be computed with relative ease. It taes a special ind of series to be telescoping, so they are fairly rare. Not all series are telescoping. Not all common series are telescoping. Don t loo for them everywhere! But don t miss them when you see them. Common errors to avoid Learning questions for Section S - Review questions:. Describe which series are telescoping and how we can identify a series as such. Memory questions:. What is a series called if its partial sums can be simplified to a small expression whose limit can be easily computed? Infinite Series Chapter : Sequences and series Section : Telescoping series Page 3

4 Computation questions: Show that each of the series in questions -5 is telescoping and use this fact to find its sum or determine divergence. n. n 3. n n n3 n n 5. n. n n n. n n n 3 Theory questions:. What is a telescoping series? 3. Does every telescoping series converge?. Which algebraic method used for integrals is used also for telescoping series? Proof questions:. Ased to compute the sum of the series n n n, a student presented the following procedure: n n n n n n Explain why this procedure is incorrect and present the proper way to obtain the conclusion, which is still! Infinite Series Chapter : Sequences and series Section : Telescoping series Page

5 . This procedure seems to prove that n x x x x x x x n n n n n n 0, something that cannot be true, since each term of the series is positive. Find the error in the procedure. x x x x x x x x x x x x x x n x n x xx 0 n x n x x nx x Templated questions:. Construct a simple telescoping series and determine its sum, if it converges. What questions do you have for your instructor? Infinite Series Chapter : Sequences and series Section : Telescoping series Page 5

6 Infinite Series Chapter : Sequences and series Section : Telescoping series Page 6