For my question, I am not looking for the solution for free, but more assistance on where I should even

*begin*. The problem and the final equation are both given in the lecture notes; unfortunately, the gap between of how to get to the solution was not provided.

Without further ado:

problem

problem

I could really use a tip in the right direction. I have been refreshing my mind of geometric series, power series, and Taylor series. My first inclination is to try to rewrite the series w(k) as the function it represents. I stumble here, because it does not quite resemble a Taylor series since 1/n! is not present, and if I attempt to correlate it with a geometric or power series of the form a+ax+ax^2... (origin at zero for simplicity), I cannot match the derivative coefficient in w(k) to the coefficient "a" from a power series.

In other words, I don't know how to plug in w(k) into the given equation in a way that I can then solve the integral. I've tried working backwards from the final equation. I can follow it mostly, except that I don't know where the sin function comes from. The Taylor series of a sin function has alternating positive and negative terms, which is not a feature of w(k).

It's driving me crazy. I'd appreciate any links, starting points, words of advice that can be given. Thank you very much!