Using known Maclaurin series to approximate modification of original

[phosgene]

Homework Statement

Recall that the Maclaurin series for sin(x) is \sum\frac{(-1)^{k}x^{2k+1}}{(2k + 1)!}.

Use this formula to find the Maclaurin polynomial P₅(x) for f(x)=xsin(x/2).

Homework Equations

The Attempt at a Solution

I know that to approximate sin(x/2) with the Maclaurin polynomial for sinx, I just substitute x/2 for x. But for xsinx, since the Maclaurin series is approximating sinx, can I just substitute the series for sinx so that I get x\sum\frac{(-1)^{k}x^{2k+1}}{(2k + 1)!}?

 

[Dansuer]

yes you can

 

[CompuChip]

Yes. You can make that better if you now sweep the x inside the sum. You may also want to include the summation bounds as you can sometimes simplify further by shifting them.

 

[phosgene]

Sorry, didn't see that I forgot the bounds. It's supposed to be from n=0 to infinity. Thanks for the help guys :)