Question

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

Series
ANSWERED
asked 2020-10-21
Representing functions by power series Identify the functions represented by the following power series.
\(\sum_{k=1}^\infty\frac{x^{2k}}{k}\)

Answers (1)

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 \(\sum_{k=1}^\infty\frac{x^{2k}}{k}=-\ln(1-x^2)\)
0
 
Best answer

expert advice

Need a better answer?
...