Question

How do you find the taylor series expansion of `f(x) =x/(1+x)` around x=0?

Answer 1

You know that `1/(1-x) = 1 +x + x^2 + x^3 + ...`

so, `1/(1+x) = 1 - x + x^2 - x^3 + ....`

therefore, `x/(1+x) = x - x^2 + x^3 - x^4 + ... = sum_{k=0}^\infty (-1)^(k)x^(k+1)`