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Exercise Find a power series for the function

$$f(x) = \int_0^x \frac{t^2}{1-t^4} \, dt$$

So basically what we want is to find $\displaystyle\sum_{n=0}^\infty a_n(x-c)^n$ so that

$$\displaystyle\sum_{n=0}^\infty a_n(x-c)^n = f(x) = \int_0^x \frac{t^2}{1-t^4} \, dt$$

I'm not exactly sure where to start or what to do. Can anyone provide suggestions/advice on how to get this started? Thanks in advance.

Grigor Hakobyan
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1 Answers1

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$$f(x) = \int_0^x \frac{t^2}{1-t^4} \, dt$$ Now, using the fundamental theorem of calculus $$f'(x)=\frac{x^2}{1-x^4}=\frac{x^2}{\left(x^2-1\right) \left(x^2+1\right)}=\frac{1}{2 \left(x^2+1\right)}-\frac{1}{4 (x+1)}+\frac{1}{4 (x-1)}$$ Write the series of each piece around $x=0$, recombine them and integrate termwise.

Claude Leibovici
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