I need to prove that
$$ \sum_{n=0}^{\infty} \left(\exp\left(\frac{n^2+2n}{n^2+1} \right) - e \right) $$
diverges. The solution suggests using the limit comparison test, but since we didn't
cover that in my class I was wondering if there is some other easy way to prove divergence.
Thank you for...
I recently watched the following video on youtube:
MF91: Difficulties with real numbers as infinite decimals I - YouTube
The guy in this video is a mathematical finitist and he claims that there is some problem with the foundation of modern mathematics, in particular modern analysis, I think...
Well not quite.
If you substitute the index of summation $ u=r-1 $ you have to change the lower as well as the upper bound of summation, because otherwise you change the number of summands. Therefore if you substitute $ u=r-1 $ we get:
$$ \sum_{r=1}^k \binom{k}{r}...
@ Sudharaka
Thank you very much for your explanation!
I have one question though: Doesn't Vandermonde's identity require the sum to start at r=0?
@Opalg
Thank you for your reply, that's a very interesting approach to the problem!
Well I have tried that of course:
$$ \sum_{r=1}^k \frac{k!}{r!(k-r)!} \frac{(n-k-1)!}{(r-1)!(n-k-r)!} \stackrel{?}{=} \frac{(n-1)!}{(k-1)!(n-k)!} $$
But I don't know where to go from here since I still can't sum the left hand side. I also tried to prove it by induction but I failed to prove the...
I'm having trouble proving the following identity (I don't even know if it's true):
$$\sum_{r=1}^k \binom{k}{r} \binom{n-k-1}{r-1}=\binom{n-1}{k-1}$$ $$\forall n,k \in \mathbb{N} : n>k$$
Thank you in advance for any help!
Vincent