Using the theorem about derivatives and integrals, we can compute the power series of more functions. Here are some examples.

• Since $\displaystyle\frac{1}{(1-x)^2}$ is the derivative of $\displaystyle\frac{1}{1-x}$, the power series for $\displaystyle\frac{1}{(1-x)^2}$ is $$\frac{d}{dx} \left(\sum_{n=0}^\infty x^n \right)= \sum_{n=1}^\infty n x^{n-1}.$$


• Since $\ln(1+x)$ is the integral of $\displaystyle\frac{1}{1+x}$, the power series for $\ln(1+x)$ is $$\int \left(\sum_{n=0}^\infty (-1)^n x^n\right) dx = \sum_{n=0}^\infty \frac{(-1)^n x^{n+1}}{n+1} = x - \frac{x^2}{2} + \frac{x^3}{3} + \cdots.$$


• Since $\tan^{-1}(x)$ is the integral of $\displaystyle\frac{1}{1+x^2}$, the power series for $\tan^{-1}(x)$ is $$\int \left(1-x^2+x^4-x^6+x^8-x^{10}+\ldots\right)\,dx = x - \frac{x^3}{3} + \frac{x^5}{5} - \frac{x^7}{7} + \cdots.$$