Answer
converges to $\dfrac{1}{1+x}$ 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 $\sum_{n=0}^\infty (-1)^n x^n=\sum_{n=0}^\infty (-x)^n$ which shows a convergent geometric series with $a=1$ and common ratio $r =-x$
$S=\dfrac{a}{1-r}=\dfrac{1}{1-(-x)}$
or, $=\dfrac{1}{1+x}$
Hence, the series $\sum_{n=0}^\infty (-1)^n x^n=\sum_{n=0}^\infty (-x)^n$ converges to $\dfrac{1}{1+x}$ for $|x| \lt 1$
