Question

Use the power series representation of f(x)=itle=" \frac{1}{1-5x} " alt=" \frac{1}{1-5x} " align="absmiddle" class="latex-formula"> to find a power series presentation of g(x)=$\frac{15}{1-5x}$

Answer 1

For $|5x|$, we have

$\dfrac1{1-5x}=\displaystyle\sum_{n\ge0}(5x)^n$

and so

$g(x)=\dfrac{15}{1-5x}=15\displaystyle\sum_{n\ge0}(5x)^n=3\sum_{n\ge0}5^{n+1}x^n$

which is, again, only valid for $|5x|$.