Partial fractions and sum of power series

Question

Im wondering how to split given power series with partial fractions (or somehow else). For example I have $$ \sum_{0}^{\infty} \frac{x^n}{4^n(n+1)(n+2)} $$ Should I just ignore $x^n$ and continue like this? $$ \frac{1}{(n+1)(n+2)}= \frac{A}{(n+1)}+\frac{B}{(n+2)}$$

Answer 1

$$\sum_{0}^{\infty} \frac{x^n}{4^n(n+1)(n+2)}=\sum_{0}^{\infty} \frac{(\frac{x}{4})^n}{(n+1)(n+2)}=\sum_{0}^{\infty} \frac{(\frac{x}{4})^n}{(n+1)}-\sum_{0}^{\infty} \frac{(\frac{x}{4})^n}{(n+2)}$$ $$=1+\frac{x}{4}\sum_{0}^{\infty} \frac{(\frac{x}{4})^n}{(n+2)}-\sum_{0}^{\infty} \frac{(\frac{x}{4})^n}{(n+2)}$$ $$=1+(\frac{x}{4}-1)\sum_{0}^{\infty} \frac{(\frac{x}{4})^n}{(n+2)}$$

Answer 2

Hint:

Recall that $$\frac{a}{bc}\equiv\frac{\frac ab}{c}$$ so you can rewrite this as:$$\sum\frac{(\frac x4)^n}{(x+1)(x+2)}=\sum\bigg[(\frac x4)^n(\frac{1}{n+1}-\frac{1}{n+2})\bigg]$$