Finding a function from its MacLaurin series?

[Xevrex]

Homework Statement

It's not exactly a specific homework question, but a Putnam one. It's an integral from 0 to inf of two multiplied MacLaurin (as far as I can tell) Series, and I'm trying to figure out how to convert one of them into a recognisable function. I'm really having trouble figuring it out though.

The series itself is $$(x - \frac{x^3}{2} + \frac{x^5}{(2)(4)} - \frac{x^7}{(2)(4)(6)} +\ ...)$$, and I've reduced it to a general form... sort of.

Homework Equations

The Attempt at a Solution

I figured that the series follows the general form of $$\sum_{n=0}^{\infty}\frac{(-1)^{n-1}(2n-1)!x^{2n-1}}{(2n-1)!}$$. It looks reminiscent of something like sin x, but I have no clue what deviation from that function would have to occur to produce that series.

By the way, I haven't formally learned Taylor/MacLaurin series, but I understand the general concepts of them--but if the method I'm asking for is generally taught within the unit, then I'm dreadfully sorry for wasting everyone's time. Every internet search I've done so far has yet to turn up anything, so...

 

[tiny-tim]

Welcome to PF!

Hi Xevrex!Welcome to PF!

Hint: try putting y = x²/2