Find a closed form of this power series f(x).

Question

$$f ( x ) = \sum _ { n = 0 } ^ { + \infty } \frac { ( - 1 ) ^ { n } ( 2 n + 3 ) } { n ! } x ^ { 2 n }$$ Please help to tell me the methods to find a closed form of the infinite series. Thank beforehand!!

Answer 1

Hints:

• $\displaystyle\sum_{n\ge 0}(-1)^n\frac{x^{2n}}{n!}=\mathrm e^{-x^2}$;
• $\displaystyle\sum_{n\ge 0}(-1)^n\frac{(x^{2n})'}{n!}=\frac1x\sum_{n\ge 0}(-1)^n\frac{2n\,x^{2n}}{n!}=\bigl(\mathrm e^{-x^2}\bigr)'$.