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marin [14]
3 years ago
5

Find a power series representation for the function. (Give your power series representation centered at x = 0.)f(x) = x3x2 + 1f(

x) = ∞n = 0 Determine the interval of convergence. (Enter your answer using interval notation.)
Mathematics
1 answer:
makkiz [27]3 years ago
8 0

I suppose you mean

f(x)=\dfrac{x^3}{x^2+1}

Recall that for |x|, we have

\dfrac1{1-x}=\displaystyle\sum_{n=0}^\infty x^n

Then

\dfrac1{1+x^2}=\dfrac1{1-(-x^2)}=\displaystyle\sum_{n=0}^\infty(-x^2)^n=\sum_{n=0}^\infty(-1)^nx^{2n}

which is valid for |-x^2|=|x|^2, or more simply |x|.

Finally,

f(x)=\displaystyle\frac{x^3}{x^2+1}=\sum_{n=0}^\infty(-1)^nx^{2n+3}

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