Question

How to show if `sum_(n=0)^oo (ln(n)^n)/95^n` converges or diverges by using root test?

`sum_(n=0)^oo (ln(n)^n)/95^n`

Answer 1

The series:

`sum_(n=1)^oo (lnn)^n/95^n`

is divergent.

Explanation:

Given the generic term of the sum:

`a_n = (ln(n))^n/95^n`

evaluate the limit:

`lim_(n->oo) root(n)(a_n) = lim_(n->oo) root(n)((ln(n))^n/95^n)`

`lim_(n->oo) root(n)(a_n) = lim_(n->oo)ln(n)/95 = oo`

As the limit is greater than `1` the series is not convergent, and as the limit has positive terms it is then divergent.

Note that the series must start from the index `n=1` because `ln(n)` is not defined for `n=0`