1. Related to harmonic series

OK so i missed this day in class and i've been trying to understand why a harmonic series is divergent. One of our homework problems is this:

We have seen that the harmonic series is a divergent series whose terms approach 0. Show that
$
\sum {ln(1+\frac{1}{n}})
$

Is another series with this property.

With infinite on the top and n=1 on the bottom of the sum sign. I don't know how to do that in LaTex.

My teacher gave use a hint as to how to solve the problem:

The sum separates into $ln(n+1)-ln(n)$

If anyone could show me how to solve this problem, explain harmonic series for me, or direct me to a website that might help i'd really appreciate it. Thanks ^.^

2. Originally Posted by mortalapeman
OK so i missed this day in class and i've been trying to understand why a harmonic series is divergent. One of our homework problems is this:

We have seen that the harmonic series is a divergent series whose terms approach 0. Show that
$
\sum {ln(1+\frac{1}{n}})
$

Is another series with this property.

With infinite on the top and n=1 on the bottom of the sum sign. I don't know how to do that in LaTex.

My teacher gave use a hint as to how to solve the problem:

The sum separates into $ln(n+1)-ln(n)$

If anyone could show me how to solve this problem, explain harmonic series for me, or direct me to a website that might help i'd really appreciate it. Thanks ^.^
You'll find the proof here (classic) http://en.wikipedia.org/wiki/Harmoni...s_(mathematics)

As for the second problem

$\sum_{n = 1}^\infty \ln \left( 1 + \frac{1}{n} \right)= \sum_{n = 1}^\infty \ln (n+1) - \ln n$. Stop after N terms and write some of them out

$S_N =$ $\left(\ln 2 - \ln 1 \right) + \left(\ln 3 - \ln 2 \right) + \left(\ln 4 - \ln 3 \right) + \cdots + \left(\ln N - \ln N-1 \right) + \left(\ln N+1 - \ln N \right)$

Everthing cancels except

$S_N = - \ln 1 + \ln(N+1) = \ln(N+1)$

Now as $\lim_{N \to \infty} S_N \to \infty$ so the series diverges.