# Representing functions by power series Identify the functions represented by the following power series. sum_{k=1}^inftyfrac{x^{2k}}{k}

Representing functions by power series Identify the functions represented by the following power series.
$\sum _{k=1}^{\mathrm{\infty }}\frac{{x}^{2k}}{k}$
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d2saint0
Given: A power series: $\sum _{k=1}^{\mathrm{\infty }}\frac{{x}^{2k}}{k}$
It is known that the power series of:
${e}^{x}=\sum _{n=0}^{\mathrm{\infty }}\frac{{x}^{n}}{n!}$
$\mathrm{ln}\left(1-x\right)=-\sum _{n=0}^{\mathrm{\infty }}\frac{{x}^{k}}{k}$
As the power series of $\mathrm{ln}\left(1-y\right)$ is:
$\mathrm{ln}\left(1-y\right)=-\sum _{n=0}^{\mathrm{\infty }}\frac{{y}^{k}}{k}$
So when $y={x}^{2}$
$\sum _{k=1}^{\mathrm{\infty }}\frac{{x}^{2k}}{k}=-\mathrm{ln}\left(1-{x}^{2}\right)$
Thus, the function for the power series $\sum _{k=1}^{\mathrm{\infty }}\frac{{x}^{2k}}{k}=-\mathrm{ln}\left(1-{x}^{2}\right)$
Jeffrey Jordon