I have a problem that i can't figure out.
a.) Consider the function f(x) = 1/(1+x^2). Write this functio as power series centered at 0.
b.) Now, use the first part find a power series for g(x) = arctan(x) also centered at x.
E is a summation.
I think i have found the first part but the second part is giving me problems i came out with this answer E(-x^(2n+2))/ (2n+2) but the answer is apparently E(-x^2n)/ (2n+2)
if someone could explain to me why that would be great!!
so its series is
This series is valid on (-1,1)
Note that the derivative of
Now from part a we know that
Now we can integrate the equation(There are some technical points that you may need to justify to exchange a sum and an integral) to get
Since we get