Math for Non-Geeks/Telescoping sums and series

oben Telescoping series are certain series where summands cancel against each other. This makes evaluating them particularly easy. Telescoping sums Introductory example Consider the sum Of course, we can compute all the brackets and then try to evaluate the limit when summing them up. However, there is a faster way: Some elements are identical with opposite pre-sign. Every two terms cancel against each other. So if we shift the brackets (associative law), we get This trick massively simplified evaluating the series. It works for any number of summands: This is called principle of telescoping sums: we make terms cancel against each other in a way that a long sum "collapses" to a short expression. General introduction thumb|Telescoping sum: Definition and explanation (YouTube video by the channel „MJ Education“) thumb|Telescoping sums work like collapsing a telescope thumb|class=noprint|A collapsible telescope A telescoping sum is a sum of the form . Neighbouring terms cancel, so one obtains: analogously, The name "telescoping sum" stems from collapsible telescopes, which can be pushed together from a long into a particularly short form. 1 Prove that . There is Definition and theorem telescoping sum A telescoping sum is a sum of the form or . value of a telescoping sum There is: telescoping sum Take the sum with and . We have Or, putting a in front of everything: Partial fraction decomposition Unfortunately, most of the sums which can be "telescope-collapsed" do not directly have the above form, but must be brought into it. The following is an example: The does not look like a telescoping sum: there is just one fraction. But there is a trick, which makes it a telescoping sum. For each we have: So And this is a telescoping sum. Who would have guessed that ?! :) The re-formulation has a name: it is called partial fraction decomposition. A fraction with a product in the denominator is split into a sum, where each summand has only one factor in the denominator. This trick can serve in a lot of cases for turning a sum over fractions into a telescoping sum. Telescoping series Introductory example What happens for infinitely many summands? Consider the series The partial sums of this series are telescoping sums: For all , there is: So the limit amounts to General introduction Telescoping series are series whose sequences of partial sums are telescoping sums. They have the form . Their partial sums have the form To see whether a telescoping series converges, we need to investigate whether the sequence converges. This sequence in turn converges, if and only if converges. If is the limit of , then the limit of the telescoping series amounts to If diverges, then diverges, as well and the entire telescoping series diverges. Analogously, the series converges, if we can show that converges. In that case, the limit is Definition, theorem and example telescoping series A telescoping series is a series of the form or . convergence of telescoping series The telescoping series and converge if and only if the sequence converges. In that case, the limits are and telescoping series The series diverges, since diverges. However, the series converges, since the sequence converges to . The limit of the series is Examples Example 1 Partial sums of the geometric series The aim of this exercise is to show the sum formula for geometric series without using induction. What we want to prove is for and . Show that the equivalent statement holds. For and there is Example 2 Does the series converge? If yes, determine the limit. We need a decomposition of the fraction here, if we want to make it a telescoping series. The denominator can be split in two factors, using the binomial theorems: Now, we can do a partial fraction decomposition as above: Hence, we get Example 3 Does the series converge? If yes, determine the limit. Again there is only one fraction with a product in the denominator, so we attempt partial fraction decomposition: This leads us to Be careful: This series is not a telescoping series! We have to add summands - not to subtract them. Even worse, the series does not converge at all: The sequence of partial sums is So they are greater as for a diverging harmonic series . By direct comparison, diverges as well. So partial fraction decomposition does not necessarily produce a telescoping sum, but it can be useful to determine whether a series converges or diverges. Series are sequences and vice versa In the beginning of the chapter, we have used that a series is actually nothing else than a sequence (of partial sums) Conversely, any sequence can be made a series if we write it as a telescoping series: We can write Question nein Why is there ? There is So a sequence element can be written as with The sequence can hence be interpreted as a series , where the "series" is seen identical with "sequence of partial sums", here. unten