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Use the power series for f(x) = 1/(1-x) to write the function g(x) = x^2/(2 - x^2) as a power series. Write your answer as a summation with a lower index of n = 0, and simplify the summation to a single variable.

Would really appreciate the help

1 Answer

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(x^2)/(2-x^2)=\frac{x^2}2\cdot\frac1{1-\frac{x^2}2}

Since


\frac1{1-x}=\displaystyle\sum_(n=0)^\infty x^n

for
|x|<1|, the power series for the rightmost rational expression is


\frac1{1-\frac{x^2}2}=\displaystyle\sum_(n=0)^\infty\left(\frac{x^2}2\right)^n=\sum_(n=0)^\infty (x^(2n))/(2^n)

which is applicable for
\left|\frac{x^2}2\right|<1, or
|x|<\sqrt2.

Distributing the other rational expression gives


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

I'm not sure what is meant by "simplify the summation to a single variable", but you can rewrite this as


\displaystyle\sum_(n=0)^\infty(x^(2(n+1)))/(2^(n+1))=\sum_(n=0)^\infty\left(\frac{x^2}2\right)^(n+1)

which could be interpreted as a simpler form, but that's really in the eye of the beholder...
User Danieln
by
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