Question

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

Answer 1

Answer:

Step-by-step explanation:

Given that:

$f_x = \dfrac{x^2}{a^7-x^7}$

$= \dfrac{x^2}{a^7(1-\dfrac{x^7}{a^7})}$

$= \dfrac{x^2}{a^7}\Big(1-\dfrac{x^7}{a^7} \Big)^{-1}$

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

Then, it implies that:

$\implies \dfrac{x^2}{a^7} \sum \limits ^{\infty}_{n=0} \Big(\Big(\dfrac{x}{a} \Big)^{^7} \Big)^n$

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

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

$\mathbf{= \sum \limits ^{\infty}_{n=0} \dfrac{x^{7n+2}}{a^{7n+7}} }}$