Solutions - 0240-04

(a) Notice the matching powers. Collect powers and then observe the geometric series pattern:

$$\begin{array}{c}\gg \gg \sum _{n=0}^{\infty }{(-2x)}^n\gg \gg \sum _{n=0}^{\infty }{u}^n\text{for }u=-2x\\ \\ \gg \gg \frac{1}{1-u}\gg \gg \frac{1}{1+2x}\end{array}$$

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(b)

$$\begin{array}{c}\sum _{n=0}^{\infty }{(-1)}^n\frac{x^{3n}}{n!}\gg \gg \sum _{n=0}^{\infty }\frac{{(-x^3)}^n}{n!}\\ \\ \gg \gg e^{-x^3}\end{array}$$