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

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Answer 1 · 2018-05-30T20:45:04

We can immediately rewrite as

$S = ln(1 * 2 * 3 * 4 * 5* 6 * 7 * ...)$

$S = ln(n!)$

If we take the limit of the sum as $n -> oo$ , we see that we get $oo$ . Therefore the series diverges.

Hopefully this helps!

Answer 2 · 2018-05-30T20:46:11

By the (nth-term) divergence test:

$lim_( n to oo) a_n = lim_( n to oo) ln (n) ne 0$

So the series diverges

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