Use the Radio Test to determine whether the series is convergent or divergent

Question

Σ With upper limit to infinite, down limit to n=1

the equation inside Σ is n!/n^n

im asked to find an

and evaluate an+1/an

Answer 1

Conduct the Ratio Test by writing a_n+1/a_n, equal here to (n + 1)!/(n + 1)^{(n + 1)} ÷ n!/nⁿ.

That is, a_n+1/a_n = (n + 1)!nⁿ ÷ (n + 1)^{(n + 1)}n! which goes to (n + 1)nⁿ ÷ (n + 1)^{(n + 1)} or

(n + 1)[n/(n + 1)]ⁿ/(n + 1) or [n/(n + 1)]ⁿ.

Then lim_(n→∞)[n/(n + 1)]ⁿ is less than 1 (since n/(n + 1) is less than 1 for all integers n from 1 to positive infinity) which signals convergence (to 1.879853862 by programmable calculator) under the Ratio Test.