Mathematical Analysis Fundamentals

in case when it converges.

Proof

If $\mid x\mid \ge 1$, then the series in Eq. (3.4) diverges by Corollary 3.27. If $\mid x\mid <1$, then from

$s_n={\displaystyle \sum _{k=0}^n}{x}^k=1+x+x^2+\cdots +x^n=\frac{1-x^{n+1}}{1-x}\text{,}$

it follows that ${\lim }_{n\to \infty }{s}_n=1/(1-x)$.  $\blacksquare$

Example 3.31

Theorem 3.30 allows us to convert a ...