Ok so the series $k.k!$ can be summed up by writing $k$ as $k+1-1$, breaking the terms up and then adding. But if we take the sum of this series uptil infinity, it comes out to be $-1!$ = $-1$(Everything cancels). What I really can't get a grip on is HOW is this possible. Every consecutive term increases considering that we have a factorial term in there I think it's safe to say that it increases a LOT. So shouldn't this series diverge? Something like :
$\dfrac{1}{1.2.3} + \dfrac{2}{2.3.4} \dots$ is still understandable. Each term decreases and as the no of terms approaches a bigger value, the terms become very small and the series converges. But what about this factorial series? Is this some wizard level thing like what Ramanujan came up with, proving the sum of all natural nos to be equal to $\dfrac{-1}{12}$? Or something simpler?Pls try and refrain from using anything above 12th grade.
Thanks!!
$$k\cdot k! = (k + 1 - 1)\cdot k! = (k+1)! - k!$$ $$\sum_{k=1}^n k\cdot k! = (n+1)! - 1$$ $$\sum_{k=1}^\infty k\cdot k! = \lim_{n\to\infty}\sum_{k=1}^n k\cdot k! = \infty$$ Nothing cancels when $n\to\infty$.