Convergence tests

Could someone explain something to me, It would be very helpful. As I am sure most people know I have an affinity for series, yet there are still three tests that allude me. The majority of this is the lack of solid comprehensible texts. These things are the following tests for convergence

Raabe's Test
"Suppose that $a_n>0$ and that as $n\to\infty$

$\frac{a_n}{a_{n+1}}=1+\frac{\alpha}{n}+O\bigg(\fra c{1}{n}\bigg)$

then the series is convergent if $\alpha>1$ , divergent if $\alpha<1$ and inconclusive for $\alpha=1$ "

I interpret this as if I take the ratio of the n-th and n+1-th term that if it reduces to 1 plus some value $\alpha$ over n and then some term that is bounded by $\frac{1}{n}$ then it applies. But how do you get a series into a form such as this?

Gauss's Test
"If $a_n>0$ and if as $n\to\infty$
$1+\frac{\beta}{n}+\frac{\tau_n}{n\ln(n)}$

The series converges if $\beta>1$ and diverges if $\beta\leq{1}$

This seems similar but I dont see where they get the $\frac{1}{n\ln(n)}$ and stuff if you are literally just taking the ratio of the nth and n+1th terms

Lastly

Kummers test

Kummer's Test -- from Wolfram MathWorld

I dont know what to make of that one,

Well most of the books containing these are written for grad students, so it is kind of hard for me to read them, and Mathworld is also the same way, so if anyone could provide and plain english explanations as to how to use these and what they mean, that would be great. Also, they seem to be substitutes for easier tests such as root or ratio.

Thanks so much in advance, this has been really bugging me