Tag: analysis

analysis – Evaluate $\int_0^1 \frac{\log \left( 1+x^{2+\sqrt{3}}\right)}{1+x}\mathrm dx$

The Question : 150 people think this question is useful I am trying to find a closed form for $$\int_0^1 \frac{\log \left( 1+x^{2+\sqrt{3}}\right)}{1+x}\mathrm dx = 0.094561677526995723016 \cdots$$ It seems that the answer is $$\frac{\pi^2}{12}\left( 1-\sqrt{3}\right)+\log(2) \log \left(1+\sqrt{3} \right)$$ Mathematica is unable to give a closed form for the indefinite integral. How can we prove this

analysis – Why does $1+2+3+\cdots = -\frac{1}{12}$?

The Question : 405 people think this question is useful $\displaystyle\sum_{n=1}^\infty \frac{1}{n^s}$ only converges to $\zeta(s)$ if $\text{Re}(s) > 1$. Why should analytically continuing to $\zeta(-1)$ give the right answer? The Question Comments : It’s not clear (to me) what you mean by “the right answer”. Since this was migrated from physics, I suspect that

geometry – The staircase paradox, or why $\pi\ne4$

The Question : 771 people think this question is useful What is wrong with this proof? Next to 180 it says “deg” So a whole side is of length $2\cdot \left(\frac{1}{2}\tan{\frac{180^\circ}{n}}\right)$ There are $n$ sides, so it has perimeter ${n\tan{\frac{180^\circ}{n}}}$ The folding method (details at the end of the answer) preserves perimeter. So the polygon’s

alculus – Under what condition we can interchange order of a limit and a summation?

The Question : 52 people think this question is useful Suppose f(m,n) is a double sequence in $\mathbb R$. Under what condition do we have $\lim\limits_{n\to\infty}\sum\limits_{m=1}^\infty f(m,n)=\sum\limits_{m=1}^\infty \lim\limits_{n\to\infty} f(m,n)$? Thanks! The Question Comments : see jstor.org/stable/1967602?seq=1#page_scan_tab_contents see en.wikipedia.org/wiki/Tannery%27s_theorem The Answer 1 56 people think this answer is useful A fairly general set of conditions, sufficient