I watched an online video lecture by some professor and she was solving a convergence problem of the power series $$\sum_{n=1}^\infty n!x^n,$$ i.e., she was finding the values of $x$ for which this power series is convergent.
She did the ratio test and winded up with $(n+1)x$ and now she started to compute the limit of this thing as $n$ approaches infinity and that's where my confusion started!
She said that :
i) If $x \neq 0$, the limit is infinity (I agree with that).
ii) If $x = 0$, the limit is $0$ (this is what I don't agree with because if $x = 0$, and $n$ approaches infinity, I should have the indeterminate form of $0\cdot\infty$. So why did she decide to make it zero?
P.S. Here is the video I'm talking about and this problem starts approximately after 6 min
https://www.youtube.com/watch?v=M8cojIKoxJg
I'd love if I can have this confusion sorted out. Thanks!
Best Answer
$\sum_{n=1}^{\infty}a_{n}$ is formally the limit $\lim_{n\rightarrow\infty}s_{n}$ where $s_{n}=\sum_{k=1}^{n}a_{k}$.
In the case you mention ($x=0$) we have $s_{n}=0$ for each $n$, hence $\lim_{n\rightarrow\infty}s_{n}=0$