Question

Write the first four nonzero terms and the general term of the Taylor serires for f about x=0. f(x)=2x/(1+x^2)

I feel that I'm doing this wrong because finding the derivatives of f(x) look really messy. Can you please show me the steps?

Thanks in advance!

Answer 1

The derivatives are messy.

f = 2x/(1+x^2)
f' = -2(x^2 - 1)/(1+x^2)^2
f'' = 4x(x^2 - 3)/(1+x^2)^3
f''' = -12(x^4 - 6x^2 + 1)/(1+x^2)^4
...