How to find the 4th and 5th coefficients ($a_4 $and $a_5$) of the serie expansion of cos(1-cos(x))
The 1 cancel with the 1 of the taylor serie of the 2nd cosine, now I have
cos( x^2/(2!)-x^4/(4!)+..).
What am I supposed to do next ? Take the taylor serie of this entire thing, that would be a mess, don’t you think ?
Thanks for your help
I think that's exactly what you're supposed to do. Note, however, that you don't need any more than $\frac{x^2}2-\frac{x^4}{24}$, as any higher order terms will clearly only contribute to $a_6$ and higher. As you're expanding $\cos\left(\frac{x^2}2-\frac{x^4}{24}\right)$ you can also rememeber that you don't need the terms with higher degree than $5$, which means you can discard a lot of terms as you apply the binomial theorem.
It's less of a mess than you think, I think.