My work and the question along with a solution is attached. My work starts at #5 written in blue. I am also attaching the Cauchy Product Theorem from my book.

I noticed that e^x has x^n for the x part whereas the sin(x) part has x^(2n+1) for the x part so I think that that is where my issue is but I don't know what to do about it.

I'm supposed to get c_1 = 1 (which is the a_1 from method 1) but I get 5/6 instead.

Any help in figuring out how to do #5 using "Method 2" would be greatly appreciated!

Thanks in advance!