Question

Find a power series representation for the function. (Assume a > 0.) (Give your power series representation centered at x = 0.) f(x) = x5 a6 − x6

Answer 1

Answer:

The answer is "$a_n =(\frac{x^{8n+5}}{a^{8n+8}})$"

Step-by-step explanation:

$f(x) =\frac{x^5}{a^8 -x^8} = \frac{\frac{x^5}{a^8}}{1-(\frac{x}{a})^8}\\\\\\\to \frac{1}{1-(\frac{x}{a})^8}$

The above equation can be writtern as the sum of the $\infty$ geometric series with $r= \frac{x}{a}$

$= \frac{\frac{x^5}{a^8}}{1+(\frac{x}{a})^8} \\\\ = \frac{x^5}{a^8} [ 1+ (\frac{x}{a})^8 + ((\frac{x}{a})^8)^2+.......]\\\\= \frac{x^5}{a^8} \sum_{\infty}^{n=0} (\frac{x}{a})^{8n}\\\\ = \sum_{\infty}^{n=0} (\frac{x^{8n+5}}{a^{8n+8}})\\\\a_n =(\frac{x^{8n+5}}{a^{8n+8}})$