Determine the X values for which the power series of n=0 to infinity N!X^N converge and the power series of n=0 to infinity X^N/N!

Step-by-step 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.

- 1.613 = -1.6 or -1.61

-1.522 = -1.5 or -1.52

0.587 = 0.6 or 0.59

3.079 = 3.1 or 3.08