# Tagged Questions

114 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 ...
185 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 ...
597 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?
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 ...