how do I find the summation notation for this power series

[calcnoob145]

this is the answer given



it says turn each term into the power series form



but I've tried it, I do not know if I'm doing it the right way usually I have a pic showing my work but my phone died if someone can guide me how to get the answer given right now I'm just guessing the answer in the form of [(-1)^k+ something*x^2k+something]/(k+something)^exponent of something but I want to do it the correct way.

 

[DrPhil]

View attachment 2781

this is the answer given

View attachment 2782

it says turn each term into the power series form

View attachment 2783

but I've tried it, I do not know if I'm doing it the right way usually I have a pic showing my work but my phone died if someone can guide me how to get the answer given right now I'm just guessing the answer in the form of [(-1)^k+ something*x^2k+something]/(k+something)^exponent of something but I want to do it the correct way.

First observation: there are only odd powers of x, --> x^(2k+1), for k=0,1,2,...

2nd observation: signs alternate --> (-1)^k

3rd observation: denominators are squares of integers, 1/(k+1)^2

That completes the verification of the form of the terms of the expansion about x=0, that is, as powers of x.

The next part of the question seems to be to use the expansion about zero to derive a power-series expansion about x=a. Do you agree with that interpretation? How would you proceed? [Note that there will be both even and odd powers.]