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Bergen

Answered question

2020-10-21

Representing functions by power series Identify the functions represented by the following power series.

\sum_{k = 1}^{\infty} \frac{x^{2 k}}{k}

Answer & Explanation

d2saint0

Skilled2020-10-22Added 89 answers

Given: A power series: $\sum_{k = 1}^{\infty} \frac{x^{2 k}}{k}$
Known properties of 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 $y = x^{2}$
$\sum_{k = 1}^{\infty} \frac{x^{2 k}}{k} = - \ln (1 - x^{2})$
Thus, the function for the power series $\sum_{k = 1}^{\infty} \frac{x^{2 k}}{k} = - \ln (1 - x^{2})$

Jeffrey Jordon

Expert2021-12-16Added 2605 answers

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