Write power series as rational function

Question

I need to write the power series: $\sum_{n=1}^\infty \frac{1}{(x-3)^{2n-1}} - \frac{1}{(x-2)^{2n-1}}$

I need to write it as a rational function. I am not sure how to go about doing this.

Answer 1

Hint:

This series has the form $$\sum_{n=1}^\infty\frac1{u^{2n-1}}-\sum_{n=1}^\infty\frac1{v^{2n-1}}=\frac1u\sum_{n=0}^\infty\frac1{(u^2)^n}-\frac1v\sum_{n=0}^\infty\frac1{(v^2)^n}.$$ Can you take it from here?

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Answer 2

In general $\sum_{n=1}^\infty\frac{1}{u^{2n-1}}=\frac{u}{u^2-1}$. Your expression becomes $\frac{x-3}{(x-3)^2-1}-\frac{x-2}{(x-2)^2-1}$$=\frac{(x-3)((x-2)^2-1)-(x-2)((x-3)^2-1)}{((x-3)^2-1)((x-2)^2-1)}$

You could simplify it. Note that $|x-3|\gt 1$ and $|x-2| \gt 1$ required.