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Please suggest a most simple sequence with the following properties:

$\sum_{n=1}^{\infty} a_n=1$

$\frac1{a_n} \sim n!$

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    At least it should mean that, according to standard notation. @Annix: What exactly $\sim$ means here?2011-04-10

3 Answers 3

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Let $a_n = 1/(n!)$ for $n \geq 2$. Then $\sum_{n=2}^\infty {a_n}$ converges to something, call the sum $L$. Let $a_1 = 1-L$. Then $\sum_{n=1}^\infty a_n = 1$.

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    In this case $L=e-2$ and $a_1=3-e \approx 0.281718\ldots$2011-04-10
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Here's an example with all the $a_n$ rational.

$ \sum_{n=1}^\infty \frac{n}{(n+1)!} = 1.$

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    @Douglas: It's many years ago now, but I first came across it as an undergraduate while answering some analysis questions (I remember thinking to myself, that's neat). It crops up from time to time on MSE, see for example http://math.stackexchange.com/questions/11665/rational-numbers-and-uniqueness/12982#129822011-04-11
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$a_n=1/(n!\sum_{n=1}^\infty 1/n!)$
(if $\sim$ means proportional)