Explanation of Cauchy's root test / criterion I've been studying some general stuff in convergence and I'm struggling with Cauchy's criterion for convergence of an infinite series. I've read in textbooks that it suggests that terms in their series should "cluster", but I'm having difficulty following the statement: "For each $\epsilon$ there is a fixed number N such that $|s_j-s_i|<\epsilon$ for all $i,j>N$ where $s_a=$a finite partial sum in the infinite series."
This general description is in all the literature and I can't follow it. I'm sure it's straightforward but I can't see it. 
On the subject of Cauchy, I'm not doing any better with Cauchy's root test, as it seems to be used for similar purposes as Cauchy's criterion, but structuraly I can't see the similarities between their structure.
Thanks for your time.
 A: A Cauchy sequence is a sequence $(s_k)$ where for any $\epsilon\gt0$, there is an $n_\epsilon$ so that for any $j,k\ge n_\epsilon$, $|s_j-s_k|\le\epsilon$.
This means that no matter how small a positive distance we want to choose ($\epsilon\gt0$), all of the remaining terms after a given point ($j,k\ge n_\epsilon$) are within that distance of each other ($|s_j-s_k|\le\epsilon$).
A sequence $(s_k)$ converges to a limit $L$ when for any $\epsilon\gt0$, there is an $n_\epsilon$ so that for all $k\ge n_\epsilon$, we have $|s_k-L|\le\epsilon$.  That is, no matter how small a distance ($\epsilon\gt0$) we choose, there is a point ($L$) so that all the remaining terms after a given point ($k\ge n_\epsilon$) are within the given distance of the limit point ($|s_k-L|\le\epsilon$).  The triangle inequality guarantees that all convergent sequences are Cauchy sequences. Furthermore, in a complete metric space, a Cauchy sequence always converges to some limit $L$. Therefore, in a complete metric space (e.g. $\mathbb{R}^n$) a sequence converges if and only if it is Cauchy. Similarly, the Cauchy Criterion Test says that, in a complete metric space, a series converges if and only if the partial sums of the series form a Cauchy sequence.
A: What you call "Cauchy's criterion" is totally different to the root test. I'll try to explain both.
Cauchy's criterion for infinite series is really just his criterion for sequences, which is that a sequence converges if (and only if, but that's a separate matter) it's a Cauchy sequence. Being a Cauchy sequence means that for any distance $\epsilon$, there's a point beyond which all terms in the sequence are within $\epsilon$ of one another. In all its symbolic glory, this is the statement:
$$(\forall \epsilon > 0\ \exists N\ \forall i, j > N)\ |a_i - a_j| < \epsilon$$
Where $(a_i)$ is just a sequence, not the terms of a series or necessarily a sequence of partial sums. If you apply this to the sequence of partial sums of a series, though, then the differences $|S_i-S_j|$ become $|x_i + ... + x_j|$, where the $x_i$ are the terms of the series. If you want to understand this test, I suggest you focus on understand the more general test for sequences, rather than series.
For the root test, you need to already understand why a geometric series converges if and only if the common ratio has absolute value less than $1$. The idea of the root test is just to compare the series to a geometric series.
A: One very useful fact of the Cauchy criterion for convergence of sequences is that it eliminates the need to mention the limit. If you want, it is an "elimination of the quantifier" $\exists$.
In words, if the terms of a sequence get closer together as the indexes get larger then the terms get closer to some number, the limit.
For series, this provides a very useful sufficient conditions of convergence or divergence (the comparison test). 
Let  series $\sum a_n$ and $\sum b_n$ be series so that  $|a_n| \le C\cdot b_n$ for some constant $C>0$ and all $n \ge $ sone $n_0$. If $\sum b_n$ is convergent then $\sum a_n$ is convergent. The equivalent contrapositive: If $\sum a_n$ is divergent then $\sum b_n$ is divergent. 
The root test, the ration test, the limit test and other tests for convergence of series are all based on the general comparison test. The comparison test itself follows from the Cauchy criterion of convergence of sequences. For the root test : the geometric series $\sum q^n$  will be the $\sum b_n$ if $0 \le q\le 1$ and $\sum a_n$ if $q \ge 1$.
