Question

Find a power series representation for the function. (Assume a>0. Give your power series representation centered at x=0 .)

f(x)=x2a7−x7

Answer 1

Explanation:

Given that:

`f_x = (x^2)/(a^7-x^7)`

`= (x^2)/(a^7(1-(x^7)/(a^7)))`

`= (x^2)/(a^7)\Big(1-(x^7)/(a^7) \Big)^(-1)`

since
`\Big((1-x)^(-1)= 1+x+x^2+x^3+...=\sum \limits ^(\infty)_(n=0)x^n\Big)`

Then, it implies that:

`\implies (x^2)/(a^7) \sum \limits ^(\infty)_(n=0) \Big(\Big((x)/(a) \Big)^(^7) \Big)^n`

`= (x^2)/(a^7) \sum \limits ^(\infty)_(n=0) \Big((x)/(a) \Big)^{^(7n)}`

`= (x^2)/(a^7) \sum \limits ^(\infty)_(n=0) \Big((x^(7n))/(a^(7n)) \Big)}`

`\mathbf{= \sum \limits ^(\infty)_(n=0) (x^(7n+2))/(a^(7n+7)) }}`