Please suggest a most simple sequence with the following properties:
$\sum_{n=1}^{\infty} a_n=1$
$\frac1{a_n} \sim n!$
Please suggest a most simple sequence with the following properties:
$\sum_{n=1}^{\infty} a_n=1$
$\frac1{a_n} \sim n!$
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 $\sum_{n=1}^\infty a_n = 1$.
Here's an example with all the $a_n$ rational.
$ \sum_{n=1}^\infty \frac{n}{(n+1)!} = 1.$
$a_n=1/(n!\sum_{n=1}^\infty 1/n!)$
(if $\sim$ means proportional)