Question

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

Answer 1

`(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...