Use the ratio or root test to check if the series converges \sum_(n=2)^(\infty) (n)/((ln(n))^n)

Question

Use the ratio or root test to check if the series converges


`\sum_(n=2)^(\infty) (n)/((ln(n))^n)`

Answer 1

Converges

Explanation:

n = k+1

(k+1) ÷ [ln(k+1)]^(k+1)

n = k

k ÷ [ln(k)]^k

{(k+1) ÷ [ln(k+1)]^(k+1)} ÷ {k ÷ [ln(k)]^k}

= {(k+1)/k} × {[ln(k)]^k}/{[ln(k+1)]^(k+1)}

{[ln(k)]^k}/{[ln(k+1)]^(k+1)} is decreasing at a much faster rate than the rate of increase of {(k+1)/k}.

Therefore the product is less than 1.

Hence it converges