Representing Functions as Power Series

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Representing Functions as Power Series

Functions as Power Series
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Applications and Examples

Using term-by-term differentiation and integration, we can compute the power series of more functions, as in the following examples.

Example 1: Since

$\displaystyle\frac{1}{(1-x)^2}$ is the derivative of

$\displaystyle\frac{1}{1-x}$ , we have

$\displaystyle\frac{1}{(1-x)^2}=\frac{d}{dx} \left(\sum_{n=0}^\infty x^n \right)= \sum_{n=1}^\infty n\, x^{n-1}.$

Example 2: Since

$\ln(1+x)$ is the integral of

$\displaystyle\frac{1}{1+x}$ , we have

$\ln(1+x)=\int \left(\sum_{n=0}^\infty (-1)^n x^n\right)\, dx = \sum_{n=0}^\infty \frac{(-1)^n \,x^{n+1}}{n+1} = x - \frac{x^2}{2} + \frac{x^3}{3} + \ldots.$

Example 3: Since

$\tan^{-1}(x)$ is the integral of

$\displaystyle\frac{1}{1+x^2}$ , we have

$\tan^{-1}(x)=\int \left(1-x^2+x^4-x^6+x^8-x^{10}+\ldots\right)\,dx = x - \frac{x^3}{3} + \frac{x^5}{5} - \frac{x^7}{7} + \ldots.$