Question

Compute
\[\lim_{n\to \infty}\frac{1\cdot 1!+2\cdot 2!+\cdots+n\cdot n!}{(n+1)!}\]

Answer 1

This looks suspiciously like a power series after you've taken its derivative and plugged in 1.

Answer 2

It's clearly an infinite series.

Answer 3

\[
\sum_{k=1}^{n}\frac{k\cdot k!}{(n+1)!}
\]

Answer 4

\[\begin{align}\lim_{n\to \infty}\frac{1\cdot 1!+2\cdot 2!+\cdots+n\cdot n!}{(n+1)!} &= \lim_{n\to \infty}\frac{\sum\limits_{n=1}^n n\cdot n!}{(n+1)!} \\~\\&=\lim_{n\to \infty}\frac{\sum\limits_{n=1}^n ((n+1)! - n! )}{(n+1)!} \\~\\& = \lim_{n\to \infty}\frac{(n+1)! - 1}{(n+1)!} \\~\\&=\cdots \end{align}\]

Answer 5

great ganeshie8

Answer 6

Yeah I gotta remember to use that trick to increase the factorial by 1 which also magically turns it into a telescoping series. Awesome haha.

Answer 7

should be :\[\begin{align}\lim_{n\to \infty}\frac{1\cdot 1!+2\cdot 2!+\cdots+n\cdot n!}{(n+1)!} &= \lim_{n\to \infty}\frac{\sum\limits_{\color{red}{k}=1}^n k\cdot k!}{(n+1)!} \\~\\&=\lim_{n\to \infty}\frac{\sum\limits_{k=1}^n ((k+1)! - k! )}{(n+1)!} \\~\\& = \lim_{n\to \infty}\frac{(n+1)! - 1}{(n+1)!} \\~\\&=\cdots \end{align}\]\]

Answer 8

Do they make you memorize these things or something?

Answer 9

lol i remember seeing the exact same/similar problem n MSE couple of days ago

Answer 10

No, @ganeshie8 is just that clever! These are fun!