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Questions tagged [ramanujan-summation]

Ramanujan summation is a technique invented by the mathematician Srinivasa Ramanujan for assigning a value to infinite divergent series.

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Help me find Taylor series for the function $ \frac{1}{(-t;q)^2 _ \infty } $ [on hold]

$$ \frac{1}{(-t;q)^2 _ \infty } $$ where $ \frac{1}{(-t;q)_ \infty } $ q-pochhammer function
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Convergence of series and summation methods for divergent series

I would like to know what is the sum of this series: $$\sum_{k=1}^\infty \frac{1}{1-(-1)^\frac{n}{k}}$$ with $$ n=1, 2, 3, ...$$ In case the previous series is not convergent, I would like to know ...
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If $(a,k)=(b,m)=1$, prove that $(ab,km)=(a,m)(b,k)$.

I'm reading the proof of the multiplicative property of $$s_k(n)=\sum_{d|(n,k)}f(d)g\bigg( \frac kd\bigg)$$ The book wrote that in order to understand the proof, we need to know if $a,b,k,m$ are ...
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How to Evaluate $1-5(\frac{1}{2})^3+9(\frac{(1)(3)}{(2)(4)})^3-13(\frac{(1)(3)(5)}{(2)(4)(6)})^3+…$

I want to Evaluate $1-5(\frac{1}{2})^3+9(\frac{(1)(3)}{(2)(4)})^3-13(\frac{(1)(3)(5)}{(2)(4)(6)})^3+...$ ,I tried from arcsin(x) series and got $\frac{1-z^4}{(1+z^4)^{\frac{2}{3}}}= 1-5(\frac{1}{2})z^...
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I want to learn about -1/12 [duplicate]

I am completely fascinated by the Euler/Ramanujan result $$ \sum_{n=0}^{\infty} n = -\frac{1}{12}$$ It is amazing to me that there are so many seemingly bogus ways to evaluate this, and they all ...
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Ramanujan summation Series [duplicate]

I am a class 12 student and I am Wondering about the Ramanujan summation Series that is 1+2+3+..... = -1/12. As per my knowledge Summation of positive number gives a positive number but in the ...
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Proof That $\sum_{k=1}^{n-1}\int_{k}^{k+1}\left\{x\right\}f´(x)dx=\int_{1}^{n}\left\{x\right\}f´(x)dx$

I am Reading the following notes: Ramanujan summation of divergent series by B Candelpergher (https://hal.univ-cotedazur.fr/hal-01150208v2/document). There, the author derives the Euler-MacLaurin ...
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Vandermonde infinite matrix inverse

I am searching for an inverse of a certain infinite matrix, Vandermonde one. I have been searching in bibliography and some well known examples exist in literature: Pascal Matrix Inverse -> ...
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Which Ramanujan's formula used the biggest constants?

After the first time I saw the movie "The Man Who Knew Infinity", about Srinivasa Ramanujan, I've looked up some of his formulas on the web. One of such formulas amazed me the most, because it used ...
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How do I derive the Ramanujan Summation of $\sum_{n=1}^{\infty}n^2 = 0$?

I'm sure everyone has seen the infamous identity of $\sum_{n=1}^{\infty}n^k=\frac{-1}{12}$, when $k=1$, and likely the associated series manipulations used to get that. I'm attempting to do a similar ...
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expanding $\prod_{n=1}^{\infty}\frac{(1-x^n)(1-x^{adn})}{(1-x^{dn})}$

I'd like to know if the infinite product can be expanded as an infinite sum: $\prod_{n=1}^{\infty}\frac{(1-x^n)(1-x^{adn})}{(1-x^{dn})}$ where $a$ and $d$ are natural numbers, thanks. Actuelly this ...
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The most complex formula for the golden ratio $\varphi$ that I have ever seen. How was it achieved?

I am fascinated by the following formula for the golden ratio $\varphi$: $$\Large\varphi = \frac{\sqrt{5}}{1 + \left(5^{3/4}\left(\frac{\sqrt{5} - 1}{2}\right)^{5/2} - 1\right)^{1/5}} - \frac{1}{e^{2\...
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What does $1+2+3+…=-\frac{1}{12}$ mean? [duplicate]

I do understand and am able to reproduce the steps to proove that $$1+2+3+...=-\frac{1}{12}$$ as, for example, shown in the Numberphile YouTube video. I can proove it, but I can't understand it. My ...
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Does Ramanujan summation extend Abel summation?

I was under the impression that if a series is Abel summable, then it is Ramanujan summable to the same value, but when I answered this question and was asked this as a follow-up, I was unable to ...
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Question on Ramanujan Summation

I know that Ramanujan Summation is used to assign a value to divergent series. Such as $$\mathfrak R\biggl(\sum_{n=0}^\infty (-1)^n\biggr)=\frac{1}{2}$$ and $$\mathfrak R\biggl(\sum_{n=1}^\infty (-1)...
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Why does $1+2+3+4+\dots=-\frac1{12}$ in a couple different ways?

$1+2+3+4+\dots$ is undefined when using regular summation. If you use either Ramanujan summation or Zeta function regularization, then $1+2+3+4+\dots=-\frac1{12}$. This article lists some over ...
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Interesting multiplicative Ramanujan-like q-expansions

We all know the full modular (cusp) form of weight 12 $$ \Delta(z) = \sum_{n=1} \tau(n)q^n = q \prod_{n=1} (1-q^n)^{24} $$ that generates the multiplicative Ramunujan tau function $\tau(n)$. Today I ...
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Infinite sum of cosecant, $\frac k{\sin(a+ck)}$

My question Does S converge or diverge? If it converged, ideally I intend to find its explicite expression as a function of c. $S=\sum_{k=1}^{\infty}\frac{k}{\sin(a+ck)}$ where $c\in R^+$ ......
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How does one get that $1^3+2^3+3^3+4^3+\cdots=\frac{1}{120}$? [duplicate]

While watching interesting mathematics videos, I found one of the papers of Srinivasa Ramanujan to G.H.Hardy in which he had written $1^3+2^3+3^3+4^3+\cdots=\frac{1}{120}$. The problem is that every ...
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Relations between Pascal's Triangle and the Zeta Function

To begin my name's Connor Evans and I'm currently an undergraduate student from Heriot Watt University in Edinburgh, in my third year of study. I have no formal training in analytic number theory but ...
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$1+1+1+1+\cdots = -\frac{21}{24}$ [closed]

In this video of numberphile they show a series of calculations to show that the sum of natural numbers is -1/12. Then I tried to proof other things using these mathematics techniques and I found the ...
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number coefficients of an infinite root of 2

Today I stumbled upon one of Ramanujan's infinite roots with all the integers and that got me curious so I started to try to create one of my own. I started with $2=2$. Then $$2=\sqrt4$$ Then $$2=\...
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Ramanujan sum. What is it, and how do we calculate it?

Can someone please explain the concept of a Ramanujan sum in easier language than Wikipedia and its relation to this question. Then, how to calculate the Ramanujan sum: $$\sum _{n\geq 1}^{\Re } n^{-1/...
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Significance of the Ramanujan summation

I wanted to know how the Ramanujan series works only using basic calculus. Why is it a shocking fact for the sum of an infinite series to be $-\frac1{12}$? How is it important to us and how does it ...
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Are all series in the elementary Ramanujan class R = 1 non-summable by analytic continuation of Dirichlet series?

We say that a series $\sum_{n=1}^\infty a_n$ and the corresponding power series $f(x)=\sum_{n=1}^\infty a_nx^n$ belong to the Ramanujan class $R=1$ if $g(x)=f(x)-f(x^2)$ is Abel summable at $x=1$ (...
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Solving function which results singularity during discrete summation

I am sorry if the following query is too basic or technically incorrect. Actually I am quite away from mathematics so I could not figure out how to solve this. I have the following function. $$f(\...
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What's the function that is related to 3 as the Riemann zeta function is related to 2?

For $f(x)=\sum_{n=0}^\infty a_n x^n$, a real number $R\neq 1$, $g(x)=f(x)-Rf(x^2)$ Abel summable at $x=1$, $g(1)=\lim_{x\to 1^-} g(x)$, the elementary Ramanujan sum of $f(x)$ at $x=1$ is defined by $f(...
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Is the linear subspace of elementary Ramanujan summable series closed under iteration?

For $f(x)=\sum_{n=0}^\infty a_nx^n$ and $g(x)=f(x)-Rf(x^2)$, $R\neq 1$, the series $\sum_{n=0}^\infty a_n$ belongs to the elementary Ramanujan class $R$ if $g(x)$ converges at least for $|x|<1$ and ...
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Convergence of infinite sum including cosh functions?

I am attempting to code up an equation that includes an infinite sum of cosine and hyperbolic cosine functions, namely: $$ \sum_{m=0}^{\infty} \frac{ \cos[(2m+1)\pi x/s] \cosh[(2m+1)\pi z/s] } {(2m+1)...
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Proof of this trivial Ramanujan result [closed]

The title is obviously sarcastic, and, sorry for my ignorance. Where can I find proofs for Ramanujan results like $$1-1+1-1+1+...= \frac{1}{2}$$ $$1+2+3+4+5+...=-\frac{1}{12}$$ I don't seem to ...
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Why is $1+2+3+\cdots = 0 $? [duplicate]

I had seen this result a while back in a Numberphile video: $1+2+3+\cdots = -\frac{1}{12}$ I was trying to prove the same result using a different method when I accidently proved that the sum was <...
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What is this quantity?

I wonder whether there is a closed form for $$-\sum_{k=1}^{\infty}\frac{\Delta^{k}\pi(x)}{k!}(-x)_k$$ where $\pi(x)$ is the prime-counting function and $(x)_k$ is the falling factorial. In other ...
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Finite Ramanujan expansions over a finite field

I'm wondering if we could have an analogy in finite fields. After all, the Discrete Fourier Transform (DFT) has been generalized to finite fields as well (with essentially identical properties as in ...
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Ramanujan sum of a function which diverges bot at $0$ and $1$

Wikipedia gives the formulas for the Ramanujan summation of a divergent series in the two cases of a function which has no divergence at $x=0$ and at $x=1$ but what to do with a function which is ...
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Interpretation of Ramanujan summation of infinite divergent series

I am not mathematician by any means so this question might be rather stupid. I came across this Wikipedia article on Ramanujan's summation and found this bewildering formula, $$1 + 2 + 3 + \dots = -...
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My (divergent) summation of the zetas with sets of cofactors give systematically errors of simple integer differences. What am I missing?

This is a "fiddling" in a small project of mine with which I'm concerned from time to time for three years now. I try to focus on the core of the problem, please ask if more context is needed. ...
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1answer
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Rewriting $\tau(p)\Delta(\tau)$ when $p$ is prime

$p$ is a prime, and $\tau$ is Ramanujan's tau function: $$p^{11}\Delta(p\tau)+\frac{1}{p}\sum_{k=0}^{p-1}\Delta\bigg(\frac{\tau + k}{p}\bigg)=p^{11}\Delta(\tau)+\frac{p^{12}}{p}\sum_{k=0}^{p-1}\Delta(...
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A Ramanujan-like summation: is it correct? Is it extensible?

I'm still exercising with summation-procedures which I try to make correct Ramanujan-summations. I'm looking at the (gap-)series $$ s(1/2,2) = (1/2)^1+(1/2)^{4}+(1/2)^{9}+(1/2)^{16}+(1/2)^{25}+... $$ ...
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If the set of natural numbers is closed under addition, how can we have the result that the sum of all the natural numbers to infinity is -1/12 [duplicate]

As seen here and on this wikipedia page the sum of all the natural numbers to infinity is -1/12. $\sum_{n=1}^\infty n = \frac{-1}{12}$ but the set of natural numbers is closed under addition and $\...
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What consistent rules can we use to compute sums like 1 + 2 + 3 + …?

$ \newcommand{ifelse}[3]{ \left( \begin{cases} #1\text{ if }#2 \\ #3\text{ otherwise} \end{cases} \right) } $ A recent Numberphile video on 1+2+3+... has made this question ("Why?") popular again, as ...
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Is there any mathematical or physical situations that $1+2+3+\ldots\infty=-\frac{1}{12}$ shows itself? [duplicate]

I just saw the proof that $$1+2+3+\cdots=-\frac{1}{12}$$ and my brain still hurts. I completely understood the proof and my question is NOT actually about the proof itself. At the end of the proof, ...
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Ramanujan 1918 paper

Does anybody know where I can find Ramanujan's paper from 1918 titled "On Certain Arithmetical Functions." It is referenced in wikipedia, under the Ramanujan Summation section, but I cannot find a ...
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Sum of the divisors of $n$, related to the Hardy-Littlewood circle method

Prove that $$\sigma(n) = \frac{\pi^2}{6}\,n\sum_{q = 1}^\infty q^{-2}c_q(n)$$ where $$ c_q(n) = \sum_{a = 1, (a, q) = 1}^q \exp(2\pi i an/q)$$ and $\sigma(n)$ is the sum of the divisors of $n$. ...
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Ramanujan Summation not consistent with Riemann's Zeta function?

Wikipedia states that Ramanujan sums and the Riemann Zeta function have the same values for even $k$: $$1 + 2^{2k} + 3^{2k} + \cdots = 0\ (\Re)$$ However, I don't understand how this can be true, ...