SumConvergence[(x - 3)^n/n, n]

Abs[-3 + x] <= 1 && x != 4

SumConvergence[x^n/Sqrt[n], n]

Abs[x] < 1

SumConvergence[(-1)^n/Sqrt[n], n]

True

SumConvergence[(1)^n/Sqrt[n], n]

False

SumConvergence[x^n/Sqrt[n], n]

SumConvergence[((-1)^(n - 1) x^n)/n^3, n]

Abs[x] < 1

Abs[x] <= 1

SumConvergence[(x - 3)^n/n, n]
(*  Abs[-3 + x] <= 1 && x != 4  *)

SumConvergence[x^n/Sqrt[n], n]
(*  Abs[x] < 1 || x == -1  *)

SumConvergence[((-1)^(n - 1) x^n)/n^3, n]
(*  Abs[x] < 1 || x == 1 || x == -1  *)