Explanation:
∑ₙ₌₀°° n! xⁿ
Use ratio test:
L = lim(n→∞)│aₙ₊₁ / aₙ│
L = lim(n→∞)│[(n+1)! xⁿ⁺¹] / (n! xⁿ)│
L = lim(n→∞)│[(n+1)! / n!] (xⁿ⁺¹ / xⁿ)│
L = lim(n→∞)│(n+1) x│
L = ∞
The series is divergent for all values of x.
∑ₙ₌₀°° xⁿ / n!
Use ratio test:
L = lim(n→∞)│aₙ₊₁ / aₙ│
L = lim(n→∞)│[xⁿ⁺¹ / (n+1)!] / (xⁿ / n!)│
L = lim(n→∞)│[xⁿ⁺¹ / (n+1)!] (n! / xⁿ)│
L = lim(n→∞)│(xⁿ⁺¹ / xⁿ) [n! / (n+1)!]│
L = lim(n→∞)│x / (n+1)│
L = 0
The series is convergent for all values of x.