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

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Discussion Overview

The discussion revolves around deriving the series expansion for the function cos(exp(-z)). Participants are exploring the application of Taylor series and the relationship between the series of cos(z) and exp(-z). The focus is on the theoretical aspects of series expansions and the challenges in combining them correctly.

Discussion Character

  • Exploratory, Technical explanation, Homework-related

Main Points Raised

  • One participant expresses difficulty in deriving the series for cos(exp(-z)) and shares their understanding of the sub-series for cos(z) and exp(-z).
  • Another participant suggests that applying the Taylor series in its basic form could yield the desired result.
  • A later reply questions whether the series is specifically around z = 0, highlighting a potential ambiguity in the initial problem statement.
  • One participant confirms that the series is indeed around z = 0, addressing the previous uncertainty.

Areas of Agreement / Disagreement

Participants generally agree that the series is centered around z = 0, but there is some uncertainty regarding the methods to derive the complete series for cos(exp(-z)).

Contextual Notes

There are limitations regarding the clarity of the steps needed to combine the sub-series, and the discussion does not resolve how to effectively derive the complete series from the given components.

Rat3dR
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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.
 
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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)
 
But isn't it the series near z = 0? They (You) didn't specify we are looking for the series near z = 0...
 
Yes it's the series around z=0. I'm sorry for not mentioning that before.. :)
 

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