Find Power Series and interval of convergence

Question

Find Power Series and interval of convergence of f(x) = xe^5x

Answer 1

Start with

e^x = 1 + x + x²/2! + x³/3! + ... = ∑_n=0^∞ xⁿ/n!, which converges everywhere.

Therefore,

 

x e^5x = x ∑_n=0^∞ (5x)ⁿ/n! = ∑_n=0^∞ 5ⁿ xⁿ⁺¹/n!, which also converges everywhere.