Question
Is it already known whether the $\zeta(4):=\sum_{n=1}^{\infty}1/n^4$ accelerated convergence series $(1)$, proved for instance in [1, Corollaire 5.3], could be obtained by a similar technique to the ones explained by Alf van der Poorten in [2, section 1] for $\zeta(3)$ and $\z...
Question. Is it already known whether the $\zeta(4)$ accelerated convergence series
$$\zeta(4)=\frac{36}{17}\sum_{1}^{\infty }
\frac{1}{n^{4}\binom{2n}{n}},\tag{1}$$
proved for instance in [1, Corollaire 5.3], could be obtained by a similar technique to those described in [2, §1] for $\z...
Convergence acceleration technique of the value of the function $\zeta(s)$ at $s=4$ (or the value of $\eta(s)$ at $s=4$) using creative telescoping/Zeilberger's algorithm?
Is it already kown if the $\zeta(4)$ accelerated convergence series
$$\zeta(4):=\sum_{n=1}^{\infty }\frac{1}{n^{4}}=\f...
New revision: **Convergence acceleration technique of the value of the function $\zeta(s)$ at $s=4$ (or the value of $\eta(s)$ at $s=4$) using creative telescoping/Zeilberger's algorithm?**
>Is it already kown if the $\zeta(4)$ accelerated convergence series $$\zeta(4):=\sum_{n=1}^{\infty }\frac{1}{n^{4}}=\frac{36}{17}\sum_{1}^{\infty } \frac{1}{n^{4}\binom{2n}{n}}\tag{1}$$ (proved e.g. in [1, Corollaire 5.3]) can be obtained by a similar technique to those described in [2, §1], where the defining series for $\zeta(3),\zeta(2)$ are accelerated yielding these equalities?
@MartinSleziak Thanks! For the first time I've used the meta sandbox to compose a future question on the main site. I've already thought of another modification where I would try to make it shorter and stress the (possible) connection to the Zeilberger algorithm as described in the book A = B. I have no means to execute the algorithm, which if I could might answer my own question.
This last equality can be justified as follows: (1) write \begin{equation*} X_{n,k}=\frac{(-1)^{k-1}}{k^{2}\binom{n+k}{k}\binom{n-1}{k}},\qquad D_{n,k}=\frac{(-1)^{k}}{n^{2}\binom{n+k}{k}\binom{n-1}{k}}\qquad k<n; \end{equation*} (2) notice that $X_{n,k}=D_{n,k-1}-D_{n,k}$, and (3) sum over $k$, $1\leq k\leq n-1$ \begin{equation*} \sum_{k=1}^{n-1}X_{n,k}=\sum_{k=1}^{n-1}\left( D_{n,k-1}-D_{n,k}\right) =D_{n,0}-D_{n,n-1}. \end{equation*} (4) Finally, sum over $n$, $1\leq n\leq N$, because \begin{equation*}
For instance the accelerated series \begin{equation*} \zeta (3)=\sum_{n=1}^{\infty }\frac{1}{n^{3}}=\frac{5}{2}\sum_{1}^{\infty } \frac{(-1)^{n-1}}{n^{3}\dbinom{2n}{n}} \end{equation*} can be obtained by letting $N\rightarrow \infty $ in \begin{equation*} \sum_{n=1}^{N}\frac{1}{n^{3}}-2\sum_{n=1}^{N}\frac{\left( -1\right) ^{n-1}}{n^{3}\binom{2n}{n}}=\sum_{k=1}^{N}\frac{(-1)^{k}}{2k^{3}\binom{N+k}{k}\binom{N}{k}}-\sum_{k=1}^{N}\frac{(-1)^{k}}{2k^{3}\binom{2k}{k}}. \end{equation*}
Does anybody know whether the $\zeta(4)$ accelerated series mencioned on section 10 of Alf van der poorten's paper A proof that Euler missed ..., transcribed at ega-math.narod.ru/Apery1.htm, can be obtain by similar techniques of those used in the same paper to derive the corresponding accelerated series for $\zeta(3),\zeta(2)$? Remark: there's a 1981 paper by Henri Cohen availble at numdam, where he used the Euler-Maclaurin summation to prove that accelerated formula for $\zeta(4)$.
A circular plate given by the relationship $x^2 + y^2 \leq 1$ is heated according to the spatial temperature function $T(x,y) = 2x^2 + y^2-y$. Find the hottest and coldest point on the plate using Lagrange multipliers.
I tried to solve it but u got only one point (0,1) instead of 4 points .. ...
@SimplyBeautifulArt Thanks! My intention was to see if there is a method for $\zeta(4)$ similar to the ones for $\zeta(2)$ and $\zeta(3)$, in order to get some idea how to obtain a similar expansion for $\zeta(5)$.
Question
Is it already known whether the $\zeta(4):=\sum_{n=1}^{\infty}1/n^4$ accelerated convergence series $(1)$, proved for instance in [1, Corollaire 5.3], could be obtained by a similar technique to the ones explained by Alf van der Poorten in [2, section 1] for $\zeta(3)$ and $\z...
... There's a 1981 paper by Henri Cohen availble at numdam, where he used the Euler-Maclaurin summation formula to prove the accelerated formula for $\zeta(4)$.
@robjohn Do you know wether the $\zeta(4)$ accelerated series mencioned on section 7 of Alf van der poorten's paper A proof that Euler missed ..., transcribed at ega-math.narod.ru/Apery1.htm, can be obtain by similar techniques of those used in the same paper to derive the corresponding accelerated series for $\zeta (3),\zeta(2)$?
What is (and what should be) the purpose of math.SE?
As far as I can say, various users have different views on this question.
Some users view it as a repository of knowledge.
Some users approach it as teaching opportunity.
Similar to previous, but slightly different: It could be understoo...
What is (and what should be) the purpose of math.SE?
As far as I can say, various users have different views on this question.
Some users view it as a repository of knowledge.
Some users approach it as teaching opportunity.
Similar to previous, but slightly different: It could be understoo...
What is (and what should be) the purpose of math.SE?
As far as I can say, various users have different views on this question.
Some users view it as a repository of knowledge.
Some users approach it as teaching opportunity.
Similar to previous, but slightly different: It could be understoo...
I see this site mainly as an opportunity to help other users clarifying their mathematical doubts, and being helped by others in my own doubts, in an exchanging process. Normally, as a non-mathematician and a non-native English speaker, I only answer low-level questions. By reading questions and...
@JorgeB. Acho que sim. É pena não haver mais utilizadores com reputação acima de 200 e, especialmente, não se terem atraído (mais) professores e linguistas, como pressupõe a descrição geral deste site, que se destinará a "linguists, teachers, and learners wanting to discuss the finer points of the Portuguese language".
Em frases com orações subordinadas causais, finais e temporais, lendo este post sobre Pontuação do blogue Livro de Estilo, as vírgulas utilizam-se conforme a posição dessas orações na frase. Se for no início ou no meio, levam obrigatoriamente vírgula; no fim, não. Penso, porém, que neste último c...
How many days we would in private beta? How many days will be? Or I wonder if is it something that will be determined by a superior decision (SE team, for example, will decide whether the private beta was successful)?
Quantos dias ficaremos em beta privado? Tem dias determinados ou é algo que ...
How to show the following identity holds?
$$
\displaystyle\sum_{n=1}^\infty\dfrac{2z}{z^2-n^2}=\pi\cot \pi z-\dfrac{1}{z}\qquad |z|<1
$$
Question
Is it already known whether the $\zeta(4):=\sum_{n=1}^{\infty}1/n^4$ accelerated convergence series $(1)$, proved for instance in [1, Corollaire 5.3], could be obtained by a similar technique to the ones explained by Alf van der Poorten in [2, section 1] for $\zeta(3)$ and $\z...
Convergence acceleration technique of the value of the function $\zeta(s)$ at $s=4$ (or the value of $\eta(s)$ at $s=4$) using creative telescoping/Zeilberger's algorithm?
Is it already kown if the $\zeta(4)$ accelerated convergence series
$$\zeta(4):=\sum_{n=1}^{\infty }\frac{1}{n^{4}}=\f...
A circular plate given by the relationship $x^2 + y^2 \leq 1$ is heated according to the spatial temperature function $T(x,y) = 2x^2 + y^2-y$. Find the hottest and coldest point on the plate using Lagrange multipliers.
I tried to solve it but u got only one point (0,1) instead of 4 points .. ...
What is (and what should be) the purpose of math.SE?
As far as I can say, various users have different views on this question.
Some users view it as a repository of knowledge.
Some users approach it as teaching opportunity.
Similar to previous, but slightly different: It could be understoo...
Falatório
Falatório
How many days we would in private beta? How many days will be? Or I wonder if is it something that will be determined by a superior decision (SE team, for example, will decide whether the private beta was successful)?
Quantos dias ficaremos em beta privado? Tem dias determinados ou é algo que ...