Taylor series with summation notation

[myusernameis]

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]

Do you know a Taylor series for \cos x?

 

[myusernameis]

Do you know a Taylor series for \cos x?

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

 

[Cyosis]

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

 

[benorin]

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]​