Recent content by VincentP

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...

I think that clarifies everything, thanks so much. Vincent

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