`sum_(n=2)^oo n/ln(n)` Determine the convergence or divergence of the series.

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`sum_(n=2)^oo n/(ln (n))`

To determine if the series is convergent or divergent, apply the nth-Term Test for Divergence.

It states that if the limit of `a_n` is not zero, or does not exist, then the sum diverges.

`lim_(n->oo) a_n!=0`      or      `lim_(n->oo) a_n =DNE`

`:.` `sum` `a_n`   diverges

Applying this, the limit of the term of the series as n approaches infinity is:

`lim_(n->oo) a_n`

`=lim_(n->oo) n/ln(n)`

To take the limit of this, use L’Hospital’s Rule.

`=lim_(n->oo) (1)/(1/n)`

`=lim_(n->oo) n`


Therefore, by the nth-Term Test for Divergence, the series diverges.

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