<s>[INST] Representing a function as a power series and finding the coefficients of the terms? My prblem asks me to find the coefficients of the first five terms of the power series representation of the function:
$$f(x)=\frac{10}{17+x}$$
$$f(x)=10*\frac{\frac{1}{17}}{1-\frac{-x}{17}}$$
$$\sum_{n=0}^{\infty}10*\frac{1}{17}*(\frac{-x}{17})^n$$
$$\sum_{n=0}^{\infty}\frac{10}{17}*\frac{(-1)^nx^n}{17^n}$$
First five terms are:
$$\frac{10}{17} + \frac{10}{17}*\frac{-x}{17} + \frac{10}{17}*\frac{x^2}{17^2} + \frac{10}{17}*\frac{-x^3}{17^3} + \frac{10}{17}*\frac{x^4}{17^4}$$
I entered $0$ for the first coefficient. Marked me correct.
I put $\frac{10}{17}$ for the second coefficient. Marked me correct.
I put $\frac{10}{17}$ for the third coefficient. Marked me wrong.
I'm not sure what I'm doing wrong. Looking at the series again, the coefficient should be $-\frac{10}{17}$ for the second coefficient, shouldn't it?
I tried a bunch of stuff for the second coefficient too. $0$, $-1$, $1$, $-10$, $10$, $\frac{10}{17}$, $-\frac{10}{17}$, $\frac{100}{17^2}$, $-\frac{100}{17^2}$, $\frac{10}{17^3}$, $-\frac{10}{17^3}$ (those last few are from different ways of writing the power series I tried). Nothing is working. Where am I going wrong?
Edit: to clarify a bit, the question wants me to find the coefficients of $x^0$, $x$, $x^2$, $x^3$ and $x^4$ in the series.
 [/INST] You allready found $$f(x)=\frac{10}{17} + \frac{10}{17}\frac{-x}{17} + \frac{10}{17}\frac{x^2}{17^2} + \frac{10}{17}*\frac{-x^3}{17^3} + \frac{10}{17}\frac{x^4}{17^4}+...$$
$$=\frac{10}{17} - \frac{10}{17^2}x + \frac{10}{17^3}x^2 - \frac{10}{17^4}x^3 + \frac{10}{17^5}x^5 \mp ...$$
The general coefficient $a_n$ is given by
$$a_n=(-1)^n\dfrac{10}{17^{n+1}}.$$
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