Given an infinite sum $S = \sum _{k = 1}^\infty a _k$, we want to know whether this sum is convergent to a single number.
1. Definitions of convergence
Definition 1.1 Given an infinite sequence ${a _0, a _1, a _2, …}$, we say the infinite series $S = \sum _{k = 0}^\infty a _k$ is convergent, if its $n$th partial sum $S _n = \sum _{k = 0} ^n a _k$ is a Cauchy sequence, that is, $$\forall \epsilon>0, \exists N, \forall n>N,|S-S _n|<\epsilon.$$
In this meaning, we can denote $$S = \lim _{n\rightarrow \infty} S _n$$
Since $\mathbb{R}$ with standard Euclidean metric is a Banach space (every Cauchy sequence converges), so convergent sequence is equivalenct to Cauchy sequence. Using the definition for Cauchy sequence, the convergence of series can be writen as:
Definition 1.2 (Cauchy Series) Given an infinite sequence ${a _0, a _1, a _2, …}$, we say the infinite series $S = \sum _{k = 0}^\infty a _k$ is convergent, if $$\forall \epsilon>0, \exists N, \forall m,n>N,|S _n-S _m|<\epsilon.$$
2. Sufficient conditions for convergence.
Theorem 2.1 (Comparison test) $\sum _{k = 0}^{\infty} a _k$ is convergent, if for all $n$, $a _n \ge 0$, and there exists a constant $M$, s.t. $\sum _{k = 0}^{n} a _k < M .$
Theorem 2.2 (Ratio test/Raabe’s test) $\sum _{k = 0} ^\infty a _k$, where $a _n > 0$ for all $n$, is convergent if $R = \lim _{n\rightarrow \infty} \frac{a _{n+1}}{a _n}<1$, and divergent if $R > 1$.
Theorem 2.3 (Root test) $\sum _{k = 0} ^\infty a _k$, where $a _n \ge 0$ for all $n$, is convergent if $R = {\lim\sup} _{n\rightarrow \infty} \sqrt[n]{a _n}<1$, and divergent if $R > 1$.
Theorem 2.4 (Dirichlet’s test) If {$a _n$} and {$b _n$} satisfy, $$a _{n+1} \le a _n$$ $$\lim _{n\rightarrow \infty} a _n = 0$$ $$|\sum _{k = 0}^N b _n| \le M, \forall N > 0$$ Then $\sum _{k=0}^n a _n b _n$ converge.
Theorem 2.5 (Abel’s test) If {$a _n$} and {$b _n$} satisfy, $$\sum _{k=0}^{\infty} a _n < \infty$$ $$b _n\ \text{is a monotone sequence}$$ $$b _n \le M, \forall n\ge0$$ Then $\sum _{k=0}^n a _n b _n$ converge.
3. Absolute Convergence and Conditional Convergence
Definition 3.1 Given a convergent series $\sum _{k =0}^\infty a _k < \infty$, we say it is absolutely convergent if $\sum _{k =0}^\infty |a _k| < \infty$. Otherwise we call it conditionally convergent.
The limit of absolute convergence is invariant of permutation of its items.
Theorem 3.2 Let $\sum _{k = 0}^\infty = A < \infty$ and $\sum _{k = 0} ^\infty |a _k| < \infty$, then for any permutation $\sigma$, $\sum _{k = 0}^\infty a _{\sigma(k)} = A$
But it is not true for conditional convergence. In fact, conditionally convergent series can permutated to get any real number! This is results from Riemann:
Theorem 3.3 (Riemann rearrangement theorem) Suppose $\{ a _k \} _{k = 0}^{\infty}$ is a sequence of real number, and $\sum _{k = 0}^\infty{a _k}$ conditionally converge. Then if $M$ is real number or $M = \infty$, there exists a permutation $\sigma$, such that $$\sum _{k = 0}^{\infty} a _{\sigma(k)} = M$$