David Lewis

2021-12-23

Identify the functions represented by the following power series. $\sum _{k=0}^{\mathrm{\infty }}{\left(-1\right)}^{k}\frac{{x}^{k}}{{3}^{k}}$

limacarp4

Expert

Given series
$\sum _{k=0}^{\mathrm{\infty }}{\left(-1\right)}^{k}\frac{{x}^{k}}{{3}^{k}}$
The Power series representation of ${\left(1+x\right)}^{-1}$ is,
$1-x+{x}^{2}-{x}^{3}+\dots =\sum _{k=0}^{\mathrm{\infty }}{\left(-1\right)}^{k}{x}^{k}$
Replace x by x/3 above we get the given series
$\sum _{k=0}^{\mathrm{\infty }}{\left(-1\right)}^{k}\frac{{x}^{k}}{{3}^{k}}={\left(1+\frac{x}{3}\right)}^{-1}=\frac{3}{3+x}$
Hence, the function represented by the power series
$\frac{3}{3+x}$
Result:
$\frac{3}{3+x}$

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