I was trying to do this problem in my book

find the Maclaurin Series for atan(x^2)using the table found on the previous page

The table listed the series for atan(x) and they apparently wanted you to plug x^2 in for x to use the table to create the series. I am trying to figure out why you can make this substitution - I know that it works but the reason why didn't jump out at me. So I tried to use the definition of Maclaurin series,

to figure out why it was ok, but I couldn't. So my question is, why can you substitute another function of x into the definition of a Maclaurin and/or Taylor Series?

Here is what I tried to do - I let u(x) be x^2 and tried to show the following

but I don't know enough about these series to work that out