Question

Find a power series representation for the function. (Assume a > 0.) (Give your power series representation centered at x = 0.) f(x) = x3 a6 − x6

Answer 1

Answer:

Power series representation of given f(x) is $\sum^{\infty}_{n=0}(-1)^{n}(\frac{x^{6n+3}}{a^{6n+6}})$

Step-by-step explanation:

Given that:

                    $f(x)=\frac{x^{3}}{a^{6}-x^{6}}$ --- (1)

writing it in form $f(x)=\frac{1}{1-x}$ for which power series is:

                    $\frac{1}{1-x}=\sum^{\infty}_{n=0}(-1)^{n}x^{n}$---(2)

                    $f(x)=\frac{x^{3}}{a^{6}}\frac{1}{1-\frac{x^{6}}{a^{6}}}$ --- (3)

consider:

                    $u=\frac{x^{6}}{a^{6}}$

substituting above eq. in (3)

                    $f(x)=\frac{x^{3}}{a^{6}}\frac{1}{1-u}$

Expanding$\frac{1}{1-u}$ using (2)

                    $=\frac{x^{3}}{a^{6}}\sum^{\infty}_{n=0}(-1)^{n}u^{n}$

Back-substituting value of u in above

                   $=\frac{x^{3}}{a^{6}}\sum^{\infty}_{n=0}(-1)^{n}(\frac{x^{6}}{a^{6}})^{n}\\=\sum^{\infty}_{n=0}(-1)^{n}(\frac{x^{6n+3}}{a^{6n+6}})$