Stuck with a Taylor series

So i have the f(x)=1/(x^2+5) and it says i must find the Taylor series around the point x0=0....iv tryed derivating it a couple of times hopeing i would see a pattern but it just feels wrong aproaching it this way... can someone please tell me how to aproach these type of exercices?

Re: Stuck with a Taylor series

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**purpleclover** So i have the f(x)=1/(x^2+5) and it says i must find the Taylor series around the point x0=0....iv tryed derivating it a couple of times hopeing i would see a pattern but it just feels wrong aproaching it this way... can someone please tell me how to aproach these type of exercices?

$\displaystyle \displaystyle \begin{align*} \frac{1}{5 + x^2} &= \frac{1}{5}\left(\frac{1}{1 + \frac{x^2}{5}}\right) \\ &= \frac{1}{5}\left[\frac{1}{1 - \left(-\frac{x^2}{5}\right)}\right] \end{align*}$

You should also know that $\displaystyle \displaystyle \begin{align*} \sum_{n = 0}^{\infty}a\,r^n = \frac{a}{1 - r} \textrm{ for } |r| < 1 \end{align*}$, and here $\displaystyle \displaystyle \begin{align*} a = 1 \end{align*}$ and $\displaystyle \displaystyle \begin{align*} r = -\frac{x^2}{5} \end{align*}$, so the series is

$\displaystyle \displaystyle \begin{align*} \frac{1}{5}\left[\frac{1}{1 - \left(-\frac{x^2}{5}\right)}\right] = \sum_{n = 0}^{\infty}{\left(-\frac{x^2}{5}\right)^n} \end{align*}$

where

$\displaystyle \displaystyle \begin{align*} \left|-\frac{x^2}{5}\right| &< 1 \\ \left|\frac{x^2}{5}\right| &< 1 \\ -1 < \frac{x^2}{5} &< 1 \\ -5 < x^2 &< 5 \\ x^2 &< 5 \\ |x| &< \sqrt{5} \\ -\sqrt{5} < x &< \sqrt{5} \end{align*}$

Re: Stuck with a Taylor series

Thanks again Prove it! ur a life savior