Question

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

Series
Representing functions by power series Identify the functions represented by the following power series.
$$\sum_{k=1}^\infty\frac{x^{2k}}{k}$$

2020-10-22
Given: A power series: $$\sum_{k=1}^\infty\frac{x^{2k}}{k}$$
It is known that the power series of:
$$e^x=\sum_{n=0}^\infty\frac{x^n}{n!}$$
$$\ln(1-x)=-\sum_{n=0}^\infty\frac{x^k}{k}$$
As the power series of $$\ln(1-y)$$ is:
$$\ln(1-y)=-\sum_{n=0}^\infty\frac{y^k}{k}$$
So when $$y=x^2$$
$$\sum_{k=1}^\infty\frac{x^{2k}}{k}=-\ln(1-x^2)$$
Thus, the function for the power series $$\sum_{k=1}^\infty\frac{x^{2k}}{k}=-\ln(1-x^2)$$