I think you need to prove that Arctan(x) has a convergent power series on the given domain. For example when does a function have a Taylor series?
Hi, i need help in the following question:
I need to prove that there are a_0, a_1,... in R and r>0 such that the series ∑(a_n)x^n converge when |x|< r, and such that ∑(a_n)x^n = arctanx for every |x|< r
What should i do here? Use the Taylor series of arctanx or something?
Thanks for any help, and sorry for the mess
Well the way I would think about it is to note that
.
You should know that for a geometric series , its infinite sum is provided .
I think you can see that if you write then you can see .
So is the closed form of . This is provided of course that .
So
.
So we have
for
for
for
for
for .