Answer:
Power series representation of given f(x) is [tex]\sum^{\infty}_{n=0}(-1)^{n}(\frac{x^{6n+3}}{a^{6n+6}})[/tex]
Step-by-step explanation:
Given that:
[tex]f(x)=\frac{x^{3}}{a^{6}-x^{6}}[/tex] --- (1)
writing it in form [tex]f(x)=\frac{1}{1-x}[/tex] for which power series is:
[tex]\frac{1}{1-x}=\sum^{\infty}_{n=0}(-1)^{n}x^{n}[/tex]---(2)
[tex]f(x)=\frac{x^{3}}{a^{6}}\frac{1}{1-\frac{x^{6}}{a^{6}}}[/tex] --- (3)
consider:
[tex]u=\frac{x^{6}}{a^{6}}[/tex]
substituting above eq. in (3)
[tex]f(x)=\frac{x^{3}}{a^{6}}\frac{1}{1-u}[/tex]
Expanding[tex] \frac{1}{1-u}[/tex] using (2)
[tex]=\frac{x^{3}}{a^{6}}\sum^{\infty}_{n=0}(-1)^{n}u^{n}[/tex]
Back-substituting value of u in above
[tex]=\frac{x^{3}}{a^{6}}\sum^{\infty}_{n=0}(-1)^{n}(\frac{x^{6}}{a^{6}})^{n}\\=\sum^{\infty}_{n=0}(-1)^{n}(\frac{x^{6n+3}}{a^{6n+6}})[/tex]
