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

Answer:

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.