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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

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

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

Example 3.31

Theorem 3.30 allows us to convert a ...

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