Solving Series of cos(exp(-z)) - Hi there!

[Rat3dR]

Hi there!

I'm new to the forum, in the sense that I've just registered, i have been reading the forum for years now, but this time I feel like i need to ask a question myself :P.

What I'm trying to figure out is how to get to the series of cos(exp(-z)). I know the result should be:

cos(exp(-z)) = cos(1) + z*sin(1) + (1/2)*z^2*(-sin(1) - cos(1)) + ...

I already figured out the "sub-series" of cos(z) and exp(-z) which are:

cos(z) = 1 - z^2/2! + z^4/4! - ...
exp(-z) = 1 - z + z^2/2! - z^3/3! + ...

I've tried many strategies to find the complete series from the two sub-series, but whatever i try, it just doesn't work.. Any hints? Because I'm completely stuck, while I'm probably just overlooking some easy, but essential, step.

Thanks, R.

 

[Rat3dR]

Allright, i found that if i apply the Taylor series in its basic form, i get the desired result..

( = f(a) + f'(a)/1! * (x-a) + .. etc)

 

[Sourabh N]

But isn't it the series near z = 0? They (You) didn't specify we are looking for the series near z = 0...

 

[Rat3dR]

Yes it's the series around z=0. I'm sorry for not mentioning that before.. :)