Can someone help me find the power series representation for this function?

[cmantzioros]

I'm trying to do the question attached. I got the first three answers correct knowing that the nth derivative of a function evaluated at 0 divided by n! = c_n. However, I did the same for the others and the answer is incorrect. I know that I need the power series representation of that function in order to get the radius of convergence but I don't know how to get it. Can someone help please?

 

[Dick]

Save yourself some labor and work out the power series expansion of log(1+x) first. It has a simple form. Then substitute 2x for x in that and multiply the whole thing by 2x. This is easier and less error prone than taking high order derivatives which get more and more complicated.

 

[cmantzioros]

ok thanks but how do I get the terms?

 

[Dick]

Take higher derivatives again. But this is a much easier function to deal with. Try it.