Question

Find a power series representation for the given function.

Answer 1

\[f(x) = \frac{ x }{ x^2+16 }\]

Answer 2

\[x * \frac{ 1 }{ 16+x^2 }=\frac{ x }{ 16 }*\frac{ 1 }{ 1-(-\frac{ x^2 }{ 16 }) }\]
\[\frac{ x }{ 16 }\sum_{n=0}^{\infty}(-\frac{ x^2 }{ 16 })^n\]\[\frac{ x }{ 16 }\sum_{}^{}(-1)^n(\frac{ x^2 }{ 16 })^n\]
\[\frac{ x }{ 16 }\sum_{}^{}(-1)^n \frac{ 1 }{ 16^n }x^{2n}=\sum_{n=0}^{\infty}(-1)^n \frac{ 1 }{ 16^{n+1} }x^{2n+1}\]
Is this correct?