Power Series and Interval of Convergence [closed]

Question

How could I find the power series and interval of convergence for x/(3+x^2)?

Answer 1

Clear["Global`*"]

f[x_] := x/(3 + x^2)

The coefficients of the power series are

coef[n_] = SeriesCoefficient[f[x], {x, 0, n},
   Assumptions -> n >= 0] // FullSimplify

(* 3^(-(1/2) - n/2) Sin[(n π)/2] *)

sum = Inactive[Sum][coef[n] x^n, {n, 0, Infinity}]

Verifying the series expansion,

f[x] == sum // Activate // Simplify

(* True *)

For the sum to be convergent

SumConvergence[coef[n] x^n, n]

(* Abs[x] < Sqrt[3] *)

Checking,

Assuming[Abs[x] < Sqrt[3], sum // Activate]

(* x/(3 + x^2) *)

Assuming[Abs[x] >= Sqrt[3], sum // Activate]