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

Mar 24, 2017

The series

${\sum}_{n = 0}^{\infty} \ln \left(\ln \left(n\right)\right)$

is not convergent.

#### Explanation:

A necessary condition for any series:

${\sum}_{n = 0}^{\infty} {a}_{n}$

to converge is that:

${\lim}_{n \to \infty} {a}_{n} = 0$

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

${s}_{n - 1} = {\sum}_{k = 0}^{n - 1} {a}_{k}$

we have:

$\left(1\right) {s}_{n} = {s}_{n - 1} + {a}_{n}$

Now if the series is convergent this means that:

${\lim}_{n \to \infty} {s}_{n} = L$ with $L \in \mathbb{R}$

and clearly this implies that also:

${\lim}_{n \to \infty} {s}_{n - 1} = L$

but as from $\left(1\right)$:

${\lim}_{n \to \infty} {s}_{n} = {\lim}_{n \to \infty} {s}_{n - 1} + {\lim}_{n \to \infty} {a}_{n}$

we have:

$L = L + {\lim}_{n \to \infty} {a}_{n}$

which implies:

${\lim}_{n \to \infty} {a}_{n} = 0$

Now as:

${\lim}_{n \to \infty} \ln \left(\ln \left(n\right)\right) = \infty$

the series is not convergent.