Cauchy's Convergence Criterion for Series
Jump to navigation
Jump to search
This article needs to be linked to other articles.
In particular: including category.
You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by adding these links.
This article, or a section of it, needs explaining, namely:
What domain is $\sequence {a_n}$ in?
You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it. To discuss this point in more detail, feel free to use the talk page.
In particular: including category.
You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by adding these links.
To discuss this in more detail, feel free to use the talk page.
Once you have done so, you can remove this instance of {{MissingLinks}} from the code.
Theorem
A series $\displaystyle \sum_{i \mathop = 0}^\infty a_i$ 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?
You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it. To discuss this point in more detail, feel free to use the talk page.
If you are able to explain it, then when you have done so you can remove this instance of {{Explain}} from the code.
Proof
Let:
- $\displaystyle s_n = \sum_{i \mathop = 0}^n a_i$
Then $\sequence {s_n}$ is a sequence in $\R$.
From Cauchy's Convergence Criterion, $\sequence {s_n}$ is convergent if and only if it is a Cauchy sequence.
For $m > n$ we have:
- $\size {s_m - s_n} = \size {a_{n + 1} + a_{n + 2} + \cdots + a_m}$
$\blacksquare$
