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# Dirichlet’s convergence test

Theorem. Let $\{a_{n}\}$ and $\{b_{n}\}$ be sequences of real numbers such that $\{\sum_{{i=0}}^{n}a_{i}\}$ is bounded and $\{b_{n}\}$ decreases with $0$ as limit. Then $\sum_{{n=0}}^{\infty}a_{n}b_{n}$ converges.

Proof. Let $A_{n}:=\sum_{{i=0}}^{n}a_{n}$ and let $M$ be an upper bound for $\{|A_{n}|\}$. By Abel’s lemma,

$\displaystyle\sum_{{i=m}}^{n}a_{i}b_{i}$ | $\displaystyle=$ | $\displaystyle\sum_{{i=0}}^{n}a_{i}b_{i}-\sum_{{i=0}}^{{m-1}}a_{i}b_{i}$ | ||

$\displaystyle=$ | $\displaystyle\sum_{{i=0}}^{{n-1}}A_{i}(b_{i}-b_{{i+1}})-\sum_{{i=0}}^{{m-2}}A_% {i}(b_{i}-b_{{i+1}})+A_{n}b_{n}-A_{{m-1}}b_{{m-1}}$ | |||

$\displaystyle=$ | $\displaystyle\sum_{{i=m-1}}^{{n-1}}A_{i}(b_{i}-b_{{i+1}})+A_{n}b_{n}-A_{{m-1}}% b_{{m-1}}$ | |||

$\displaystyle|\sum_{{i=m}}^{{n}}a_{i}b_{i}|$ | $\displaystyle\leq$ | $\displaystyle\sum_{{i=m-1}}^{{n-1}}|A_{i}(b_{i}-b_{{i+1}})|+|A_{n}b_{n}|+|A_{{% m-1}}b_{{m-1}}|$ | ||

$\displaystyle\leq$ | $\displaystyle M\sum_{{i=m-1}}^{{n-1}}(b_{i}-b_{{i+1}})+|A_{n}b_{n}|+|A_{{m-1}}% b_{{m-1}}|$ |

Since $\{b_{n}\}$ converges to $0$, there is an $N(\epsilon)$ such that both $\sum_{{i=m-1}}^{{n-1}}(b_{i}-b_{{i+1}})<\frac{\epsilon}{3M}$ and $b_{i}<\frac{\epsilon}{3M}$ for $m,n>N(\epsilon)$. Then, for $m,n>N(\epsilon)$, $|\sum_{{i=m}}^{n}a_{i}b_{i}|<\epsilon$ and $\sum a_{n}b_{n}$ converges.

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new question: Lorenz system by David Bankom

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new question: Latent variable by adam_reith

new question: Lorenz system by David Bankom

Oct 19

new correction: examples and OEIS sequences by fizzie

Oct 13

new correction: Define Galois correspondence by porton

Oct 7

new correction: Closure properties on languages: DCFL not closed under reversal by babou

new correction: DCFLs are not closed under reversal by petey

Oct 2

new correction: Many corrections by Smarandache

Sep 28

new question: how to contest an entry? by zorba

new question: simple question by parag

Sep 26

new question: Latent variable by adam_reith

## Comments

## questions re: Dirichlet's convergence

Should the second line be the summation from i=0 to i=n-1 and the summation from i=0 to i=m-2?

Also, in the last line, $ \vert\sum_{i=m}^n a_ib_i\vert<\epsilon$ < infinity, therefore $ \sum a_nb_n$ converges?

Thanks.

## Re: questions re: Dirichlet's convergence

Agree with first comment; the sums are not correctly indexed.

Re the last line, it is correct, but you are using the Cauchy criterion: I recommend you quote it.