2
votes
3answers
106 views

Proof without words for $\sum_{i=0}^\infty(-1)^i\frac{1}{2i+1}$

$$\sum_{i=0}^\infty(-1)^i\frac{1}{2i+1}$$ $$1-\frac13+\frac15-\frac17+\frac19-\cdots=\frac\pi4$$ Does anyone know of a proof without words for this? I am not looking a for a just any proof, since I ...
3
votes
5answers
175 views

$\sum_{k=1}^nH_k = (n+1)H_n-n$. Why?

This is motivated by my answer to this question. The Wikipedia entry on harmonic numbers gives the following identity: $$ \sum_{k=1}^nH_k=(n+1)H_n-n $$ Why is this? Note that I don't just ...
5
votes
5answers
540 views

Proving $\sum_{k=1}^n{k^2}=\frac{n(n+1)(2n+1)}{6}$ without induction [duplicate]

I was looking at: $$\sum_{k=1}^n{k^2}=\frac{n(n+1)(2n+1)}{6}$$ It's pretty easy proving the above using induction, but I was wondering what is the actual way of getting this equation?
2
votes
0answers
110 views

Proof for a summation-procedure using the matrix of Eulerian numbers?

I've discussed a procedure for divergent summation using the matrix of Eulerian numbers occasionally in the last years (initially here, and here in MSE and MO but not in that generality and thus(?) ...
4
votes
1answer
136 views

About Euler's formula for Apery number

Euler's formula. $$\zeta(3)=\frac{\pi^2}{7}\left(1-4\sum_{m\ge 1}\frac{\zeta(2m)}{(2m+1)(2m+2)2^{2m}}\right)$$ I saw this formula in Wikipedia a few months ago. I have searched about Euler's ...