One is often asked to check if a given series converges or not in one's first year of uni. Is there a formal definition that allows us to check this? We are only given a bunch of tests that are troublesome to remember (I've never liked cramming in math), and a Google-search only yields results regarding sequences.
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$\begingroup$The formal definition is that $$ \sum_{k=1}^\infty a_k $$ converges exactly if the sequence of it's partial sums converges, i.e. if the sequence $(s_n)_{n\in\mathbb{N}}$ defined by $$ s_n = \sum_{k=1}^n a_k $$ converges. In other words, you per definition have that $$ \sum_{k=1}^\infty a_k = \lim_{n\to\infty} s_n = \lim_{n\to\infty} \sum_{k=1}^n a_k $$
$\endgroup$ 5 $\begingroup$A series $a_n$converges if the limit $$\lim_{n\to \infty}\sum_{i=0}^n{a_i}=L$$ exists and is finite. In other words, for all $\epsilon>0$ there exists an $N$ such that for all $n>N$, $$\left|L-\sum_{i=0}^n{a_i}\right|<\epsilon$$
$\endgroup$ $\begingroup$Let $\sum_{n=m}^\infty a_n$ be a formal infinite series. For any integer $N\geqslant m$, we define the $N^{\text{th}}$ partial sum $S_N$ of this series to be $S_N:=\sum_{n=m}^N a_n$; of course, $S_N$ is a real number. If the sequence $(S_N)_{n=m}^\infty$ converges to some limit $L$ as $N\to\infty$, then we say that the infinite series $\sum_{n=m}^\infty a_n$ is convergent, and converges to $L$; we also write $L=\sum_{n=m}^\infty a_n$, and say that $L$ is the sum of the infinite series $\sum_{n=m}^\infty a_n$. If the partial sums $S_N$ diverge, then we say that the infinite series $\sum_{n=m}^\infty a_n$ is divergent.
from Terence Tao - Analysis I.
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