Question

How do you determine whether the sequence `a_n=ln(ln(n))` converges, if so how do you find the limit?

Answer 1

The series

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

is not convergent.

Explanation:

A necessary condition for any series:

`sum_(n=0)^oo a_n`

to converge is that:

`lim_(n->oo) a_n = 0`

In fact if we consider the `n`-th partial sum:

`s_(n-1) = sum_(k=0)^(n-1) a_k`

we have:

`(1) s_n = s_(n-1) +a_n`

Now if the series is convergent this means that:

`lim_(n->oo) s_n = L` with `L in RR`

and clearly this implies that also:

`lim_(n->oo) s_(n-1) = L`

but as from `(1)`:

`lim_(n->oo) s_n = lim_(n->oo) s_(n-1) + lim_(n->oo) a_n`

we have:

`L = L + lim_(n->oo) a_n`

which implies:

`lim_(n->oo) a_n = 0`

Now as:

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

the series is not convergent.