Proof of Raabe's test Let $\sum a_n$ be a series of non-negative terms and let $$L = \lim_{n\to\infty}n\left(1-\frac{a_{n+1}}{a_n}\right)$$
Prove that the series converges (resp. diverges) if $L > 1$ (resp. $L<1$). I've tried, for example, that when $L<1$, $$n\left(1-\frac{a_{n+1}}{a_n}-L\right)= \frac{n(a_n-a_{n+1}-La_n)}{a_n}\ge\frac{n(a_n-a_{n+1}-a_n)}{a_n}=\frac{-na_{n+1}}{a_n}$$ and then using epsilons and the sort, but I can't get anywhere. Any tips?
P.S. using Kummer's test doesn't count
 A: If $L>1$, choose $\epsilon\gt 0$, $L-\epsilon>1$ then 
$$1-\frac{L-\epsilon}{n} > \frac{a_{n+1}}{a_n}$$
Choose $p$ such that $1\lt p\lt L-\epsilon$, $\sum\frac1{n^p}$ converges. $b_n=\frac1{n^p}$,  if $n$ big enough, then
$$\frac{b_{n+1}}{b_n}=(1-\frac{1}{n+1})^p=1-\frac{p}{n}+O(\frac1{n^2})\gt 1-\frac{L-\epsilon}{n}>  \frac{a_{n+1}}{a_n}$$ 
so $\sum a_n$ converges
A: The series even converges absoute, thus set $a_n:=\left|a_n\right|$
Now (if $n$ is big enough)
$$L = \lim_{n\to\infty}n\left(1-\frac{a_{n+1}}{a_n}\right) \Longleftrightarrow \frac{a_{n+1}}{a_n}\leq\frac{n-L}{n}$$
This is equivalent to
$$\left(L-1\right)a_n \leq \left(n-1\right)a_n-na_{n+1}$$
Because of $L>1$ the left side is bigger than zero, so
$$0\leq \left(n-1\right)a_n-na_{n+1}\Longleftrightarrow na_{n+1}\leq \left(n-1\right)a_n$$
That means that $a_n$ is decreasing and bounded, so it does converges. Now define the sum
$$\sum b_n = \sum \left(n-1\right)a_n-na_{n+1}$$
Which is a telescop-sum and thus it converges
But that implies that the sum over $a_n$
$$\sum a_n \leq \sum \left(L-1\right)a_n \leq \sum b_n$$
is convergent as well by the comparison test.
A: The following is a new argument for the convergence part. Assume that $L>1$. Choose $\varepsilon > 0$ small enough such that
$$L-\varepsilon > 1.$$
There exists some $1 \ll N=N(\varepsilon)$ such that
$$n\Big( 1- \frac {a_{n+1}}{a_n} \Big) > L-\varepsilon$$
for any $n \geq N$, namely
$$\frac {a_{n+1}}{a_n} < 1 - \frac{L-\varepsilon}n = \frac{n-(L-\varepsilon)}n$$
for any $n \geq N$. Since
$L-\varepsilon>1$, one can always choose $\alpha >1$ in such a way that
$$\frac{n-(L-\varepsilon)}n < \Big( \frac{n-1}n \Big)^\alpha$$
for any $n \gg 1$, say $n \geq M > N$. Hence for large $n$, we obtain
$$ a_{n+1} \leq \Big( \frac{n-1}n \Big)^\alpha a_n \leq \Big( \frac{n-2}n \Big)^\alpha a_{n-1} \leq \cdots \leq \Big( \frac{M-1}n \Big)^\alpha a_M.$$
Hence $\sum a_n$ converges by the comparison test.
