Cauchy's Convergence Criterion for Series
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What domain is $\sequence {a_n}$ in?
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Theorem
A series $\ds \sum_{i \mathop = 0}^\infty a_i$ is convergent if and only if for every $\epsilon > 0$ there is a number $N \in \N$ such that:
- $\size {a_{n + 1} + a_{n + 2} + \cdots + a_m} < \epsilon$
holds for all $n \ge N$ and $m > n$.
What domain is $\sequence {a_n}$ in?
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Proof
Let:
- $\ds s_n = \sum_{i \mathop = 0}^n a_i$
Then $\sequence {s_n}$ is a sequence in $\R$.
From Cauchy's Convergence Criterion on Real Numbers, $\sequence {s_n}$ is convergent if and only if it is a Cauchy sequence.
For $m > n$ we have:
- $\size {s_m - s_n} = \size {a_{n + 1} + a_{n + 2} + \cdots + a_m}$
$\blacksquare$