# Taylor series with summation notation

## Homework Statement

f(x) = $$\frac{1-cos(X^2)}{x^3}$$

which identity shoud i use?
and tips on this type of questions? once i can separate them, then i'll be good

thanks!

benorin
Homework Helper
Gold Member
Do you know a Taylor series for $\cos x$?

Do you know a Taylor series for $\cos x$?

yeah, but there's a x^3 on the bottom...

Cyosis
Homework Helper
Sure, but the summation isn't over x so you can put the x in the sum or outside the sum.

benorin
Homework Helper
Gold Member
Example:

$$\frac{1-\sin 2x^3}{x}=\frac{1-\sum_{k=0}^{\infty}\frac{\left( 2x^3\right)^{2k+1}}{(2k+1)!}}{x} = {\scriptstyle \frac{1}{x}}-{\scriptstyle \frac{1}{x}}\sum_{k=0}^{\infty}\frac{2^{2k-1}x^{6k+3}}{(2k+1)!}$$
= {\scriptstyle \frac{1}{x}}-\sum_{k=0}^{\infty}\frac{2^{2k-1}x^{6k+2}}{(2k+1)!}[/tex]​