Question

Find the th partial sum of the telescoping series, and use it to determine whether the series converges or diverges. If it converges, find its sum.

Answer 1

To find the th partial sum of a telescoping series, identify the pattern of cancellation and express the th partial sum in terms of the first th term. Check if the series converges or diverges by taking the limit as n approaches infinity. If the series converges, find its sum by evaluating the limit as n approaches infinity of the th partial sum.

Step-by-step explanation:

A telescoping series is a series in which most of the terms cancel out, leaving only a few terms. To find the th partial sum of a telescoping series, we need to first identify the pattern of cancellation. Once we have the pattern, we can express the th partial sum in terms of the first th term.

After finding the th partial sum, we can check if the series converges or diverges by taking the limit as n approaches infinity. If the limit exists and is a finite number, then the series converges. If the limit is infinity or does not exist, then the series diverges.

If the series converges, we can find its sum by evaluating the limit as n approaches infinity of the th partial sum.

Answer 2

Step-by-step explanation:

1) In a telescoping series almost every term cancels either with a prior or with the subsequent term. To determine if a Series converges to a finite value or infinite value we need to calculate its partial sum.

2) So let's start with this sum:

`S=1-1+1-1+1...\sum_(n=1)^(\infty)(-1)^(n-1)\\`

3) Let's sum its partial sum to find out if it converges or diverges, i.e.:

`\\\sum_(n=1)^(N)(-1)^(n-1)=1+0+1+0+1\\n=1 \:S=1\\n=2\:S=1-1=0\\n=3\:S=1-1+1=1\\n=4\:S=1-1+1-1=0\\n=5\:S=1-1+1-1+1=1`

4) The value for this

`\\\lim_(n\rightarrow \infty)\sum_(n=1)^(N)(-1)^(n-1)=Limit\: Does\: not\:Exist`

Then this Telescoping Series is divergent, because it does not converge to a finite value.