Answer: The given series neither converges nor diverges.
Step-by-step explanation: We are given to determine whether the following series converges or diverges :
[tex]S=\sum_{n=0}^{\infty}(-1)^n.[/tex]
If the series converges, we are to find its sum.
The given series can be written as :
[tex]1,~-1,~1,~-1,~1,~1,~~.~~.~~.[/tex]
We note that the given series is a geometric one with first term 1 and common ratio given by
[tex]r=\dfrac{-1}{1}=\dfrac{1}{-1}=~~.~~.~~.~~=-1.[/tex]
We know that a geometric series with common ratio r converges if |r| <1 and diverges if |r| > 1.
Since |r| = 1 for the given series, so the series will neither converge nor diverge.
Thus, the given series neither converges nor diverges.
