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The problem: Let us define$$mathscr{F}(u)=int_0^1(u'(x))^4-e^xsin (u(x)), mathrm{d}x$$for $u in W^{1,1}([0,1])$ such that $u(0)=A$ and $u(1)=B$.It don't matters what $A$ or...
Asked on 11/09/2021
0 answerDoes there exist a formula akin to Cauchy's Repeated Integration formula, but for derivatives? Cauchy's formula doesn't seem ideal for finding, say, the 100th derivative of a function as factorials...
Asked on 11/09/2021
1 answerLemma : Suppose that the events $A_{1}, ldots, A_{n}$ satisfy $operatorname{Var}left(sum_{i=1}^{n} I_{A_{i}}right) leq c sum_{i=1}^{n} Pleft{A_{i}right} .$ Then :$$ left(1-Pleft(bigcup_{k=1}^{n} A_{k}right)right)^{2} sum_{i=1}^{n} Pleft{A_{i}right} leq c Pleft(bigcup_{k=1}^{n}...
Asked on 11/09/2021
2 answer(citation: Munkres Topology 7.5 b) Backgound (from Munkres Topology Chapter 7): Theorem 7.1: $B$ is a nonempty countable set $LeftarrowRightarrow$ there is an injective function $g:...
Asked on 11/09/2021
1 answerFrom another problem, I got stuck trying to solve this limit: $$lim_{kto +infty}prod_{v=k+1}^{2k}left(1-frac1{va}right)$$where $a>1$ is a positive integer. I tried to take $log$ on both sides...
Asked on 11/09/2021
3 answerGiven that $sum a_{n}$ converges $left(a_{n}>0right) ;$ Then $(sum a_{n}^{3} sin n)$ isMy approach: Since, $sum a_{n}$ converges, we have $lim _{n rightarrow infty}...
Asked on 11/09/2021 by user791682
2 answerWe have the following result ($text{Li}_{n}$ being the polylogarithm): $$tag{*}small{ int_0^1 log^2 (1-x) log^2 x log^3(1+x) frac{dx}{x} = -168 text{Li}_5(frac{1}{2}) zeta (3)+96 text{Li}_4(frac{1}{2}){}^2-frac{19}{15} pi ^4 text{Li}_4(frac{1}{2})+\ 12 pi...
Asked on 11/09/2021
2 answerWe have to prove that the number $$N=512^3 + 675^3 + 720^3$$ is composite.I tried to use the identity $(a^3+b^3+c^3)=(a+b+c)(a^2+b^2+c^2-ab-bc-ca)+3abc$ hoping to take...
Asked on 11/09/2021
1 answerIt is well-known that:$$lim_{kto+infty}frac{sin(kx)}{pi x}=delta(x).$$This can also be written as$$ 2pidelta(x)=int^{+infty}_{-infty}e^{ikx},mathrm dk.$$However, I don't know how to prove this without using Fourier Transform. I...
Asked on 11/09/2021 by lrh2000
4 answerThe sides of a circular track contain a sequence of cans of gasoline. The total amount in the cans is sufficient to enable a certain car to make one complete...
Asked on 11/09/2021
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