# How do you show that the series ln1+ln2+ln3+...+lnn+... diverges?

May 30, 2018

We can immediately rewrite as

$S = \ln \left(1 \cdot 2 \cdot 3 \cdot 4 \cdot 5 \cdot 6 \cdot 7 \cdot \ldots\right)$

S = ln(n!)

If we take the limit of the sum as $n \to \infty$, we see that we get $\infty$. Therefore the series diverges.

Hopefully this helps!

May 30, 2018

By the (nth-term) divergence test:

${\lim}_{n \to \infty} {a}_{n} = {\lim}_{n \to \infty} \ln \left(n\right) \ne 0$

So the series diverges