Answer
converges to $\dfrac{1}{1+x^2}$ for $|x| \lt 1$
Work Step by Step
Formula to calculate the sum of a geometric series is:
$S=\dfrac{a}{1-r}$;
Consider the series $\sum_{n=0}^\infty (-1)^n x^{2n}=\sum_{n=0}^\infty (-x^2)^n$; which shows a convergent geometric series with $a=1$ and common ratio $r =-x^2$
Also, $S=\dfrac{a}{1-r}=\dfrac{1}{1-(-x^2)}$
or, $S=\dfrac{1}{1+x^2}$
Hence, the series $\sum_{n=0}^\infty (-1)^n x^{2n}=\sum_{n=0}^\infty (-x^2)^n$; converges to $\dfrac{1}{1+x^2}$ for $|x| \lt 1$
