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# Is there a formal definition of convergence of series?

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.

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 $$\sum_{k=1}^\infty a_k = \lim_{n\to\infty} s_n = \lim_{n\to\infty} \sum_{k=1}^n a_k$$

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 $n>N$, $$\left|L-\sum_{i=0}^n{a_i}\right|<\epsilon$$

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# Is there a formal definition of convergence of series?

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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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 $$\sum_{k=1}^\infty a_k = \lim_{n\to\infty} s_n = \lim_{n\to\infty} \sum_{k=1}^n a_k$$

Oh, of course, that was simple, yet something I was completely unaware of consciously. I have to agree with @Seth, it is indeed weird that we haven't formalized it. - Andrew Thompson 2 days ago

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