Integrating infinite sums and macluarin's expansion

[rock.freak667]

Homework Statement

Using the macluarin's expansion for sinx show that \int sinx dx=-cosx+c

Homework Equations

sinx=\sum_{n=0} ^\infty \frac{(-1)^nx^{2n+1}}{(2n+1)!}

The Attempt at a Solution

Well I can easily write out some of the series and just show that it is equal to -cosx

but if I integrate the representation for the infinite series i get

\int sinx dx= \sum_{n=0} ^\infty \frac{(-1)^nx^{2n+2}}{(2n+1)!}
shouldn't -cosx be:
\int sinx dx= \sum_{n=0} ^\infty \frac{(-1)^{n+1}x^{2n+2}}{(2n+1)!}

and also I am supposed to get x^{2n} not what I got

 

[Avodyne]

Your denominators should be (2n+2)! in the last two lines.

With this correction, your last series is cos(x)-1; write out the first few terms to see.

 

[rock.freak667]

ah yes I made a typo...but even if I change it
how do I manipulate

\sum_{n=0} ^\infty \frac{(-1)^nx^{2n+2}}{(2n+2)!} (what I integrated and got)

Into

-\sum_{n=0} ^\infty \frac{(-1)^nx^{2n}}{(2n)!} (What I am supposed to get)

 

[HallsofIvy]

Let j= n+1 and see what you get with j as the index.