While no system is full-proof, including ours, we will continue using internet sum paid for an Indian modern or contemporary art sold at auction. Integration in 1997 Veer Savarkar Award in 1998 Ramanujan Award in 2000

14 Jul 2016 Our first question is to prove the following equation involving an infinite There is a certain house on the street such that the sum of all the

If I am right and the sum is actually –3/32, then we are in trouble here, because this implies that some statements of string theory are based on an incorrect result. The "proof" in general is using ramanjuan summation and analytic continuation of the riemann function. In this proof, the election of the riemann function in order to perform the analytic continuation seems just like one of the infinite functions we can choose. So the questions would be: Ramanujan Summation's -1/12 is not an element of the group of all positive integers. Does this prove the summation wrong?

Ramanujan summation proof

Proof. A proof subject to "natural" assumptions (though not the weakest necessary conditions) to Ramanujan's Master theorem was provided by G. H. Hardy employing the residue theorem and the well-known Mellin inversion theorem. Application to Bernoulli polynomials Ramanujan summation is a method to isolate the constant term in the Euler–Maclaurin formula for the partial sums of a series. For a function f , the classical Ramanujan sum of the series ∑ k = 1 ∞ f ( k ) {\displaystyle \sum _{k=1}^{\infty }f(k)} is defined as Ramanujan’s Formula for Pi. First found by Ramanujan.

So the questions would be: Ramanujan Summation's -1/12 is not an element of the group of all positive integers.

A simple proof of Ramanujan's summation of the . George E Andrews; Richard Askey. Aequationes mathematicae (1978) Volume: 18, page 333-337; ISSN: 0001-9054; 1420-8903/e; Access Full Article top Access to full text. How to cite top

Ramanujan once derived the same formula without usin 20 Feb 2018 How did the astounding autodidact Srinivasa Ramanujan achieve rigorous proofs, she added, and Ramanujan's notebooks – examples it was the smallest number expressible as a sum of two cubes in two distinct ways. 17 Jan 2014 -1/12 is called Ramanujan summation, which in turn is based on and they have another video explaining the correct proof using them. 12 Dec 2018 This prove is in this attachment.it may help you to understand Ramanujan series. 24 Jan 2014 The sum of all natural numbers is equal to -1/12.

3 Mar 2020 In this video I show you how to use mathematical induction to prove the sum of the series for ∑r. The method of induction: Start by proving that it

G. E. Andrews and R. Askey, A simple proof of Ramanujan’s summation of the 1 1, Ae-quationes Mathematicae 18 (1978), 333 Abstract We present a new proof of Ramanujan's 1 ψ 1 summation formula. You are currently offline. Some features of the site may not work correctly. Hi, i've seen several videos and documents that state that "the sum of all natural numbers is equal to -1/12". The "proof" in general is using ramanjuan summation and analytic continuation of the riemann function. In this proof, the election of the riemann function in order to perform the A simple proof by functional equations is given for Ramanujan’s 1 ψ 1 sum. Ramanujan’s sum is a useful extension of Jacobi's triple product formula, and has recently become important in the treatment of certain orthogonal polynomials defined by basic hypergeometric series.

× 13591409 + 545140134 n 640320 3 n Ramanujan Summation's -1/12 is not an element of the group of all positive integers. Does this prove the summation wrong? [duplicate] Ramanujan's Summation says that the sum of all integers is -1/12 1 + 2 + 3=-1/12. If we define group G to be group of all positive integers, then the group contains all positive integers. G.H. Hardy recorded Ramanujan’s 1 1 summation theorem in his treatise on Ramanujan’s work [17, pp. 222–223] .
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Although the Ramanujan summation of a divergent series is not a sum in the traditional sense, it has properties that make it mathematically useful in the study of divergent infinite series, for which conventional summation is undefined.

Concrete experiments are given to prove the robustness of the 2 Dec 2013 The first published proof was given by W. Hahn [1] in 1949. Theorem. ( Ramanujan's ${}_1\psi_1$ Summation Formula) If $|\beta q|< 14 Jul 2016 Our first question is to prove the following equation involving an infinite There is a certain house on the street such that the sum of all the 27 Apr 2016 The sum of all positive integers equal to -1/12 Littlewood speculated that Ramanujan might not be giving the proofs they assumed he had 14 Dec 2012 Rogers–Ramanujan and dilogarithm identities Although we prove the 5-term relation for x and y restricted to the interval (0,1), and this classical summation or transformation formula which involves positive terms i 21 Nov 2017 when s>1 and as the “analytic continuation” of that sum otherwise. A commenter pointed out that it's a pain to find a proof for why Euler's sum works.
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alternatively, a short proof of the recent result of Bradley about Ramanujan's enigmatic claim. For complex numbers α, β, γ and integer δ, define the sum of

Yup, -0.08333333333. G.H. Hardy recorded Ramanujan’s 1 1 summation theorem in his treatise on Ramanujan’s work [17, pp. 222–223] .

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What most surprised me is discovering that the Ramanujan summation is used in string theory and quantum mechanics.

20 Feb 2018 How did the astounding autodidact Srinivasa Ramanujan achieve rigorous proofs, she added, and Ramanujan's notebooks – examples it was the smallest number expressible as a sum of two cubes in two distinct ways.

Few days ago I thought about proof of :$$\frac{1}{3}+\frac{1}{3\cdot 5} + \dots = \sqrt{\frac{e\pi}{2}}$$. I tried to represent my sum as : $$\sum\frac{2n!!}{(2n+1 The proof of Hardy and Ramanujan of their formula for P(n) is complicated, and few professional mathematicians have examined and appreciated all its intricacies.

For example, for m =3 we get G. E. Andrews and R. Askey, A simple proof of Ramanujan’s summation of the 1 1, Ae-quationes Mathematicae 18 (1978), 333{337. Show, by a judicious choice of the parameters a, band x, that Ramanujan’s formula (2) implies that (1) has the product representation f(z; ;q) = 1 z (1 z)(1 ) Y1 n=1 (1 qn)2 (1 zqn)(1 z 1qn) Y1 n=1 (1 zqn)(1 ( z) 1qn) Request PDF | Proofs of Ramanujan's1ψ1i-summation formula | Ramanujan's i 1ψ1-summation formula is one of the fundamental identities in basic hypergeometric series. We review proofs of this A simple proof by functional equations is given for Ramanujan’s1 ψ 1 sum. Ramanujan’s sum is a useful extension of Jacobi's triple product formula, and has recently become important in the Proof A proof subject to "natural" assumptions (though not the weakest necessary conditions) to Ramanujan's Master theorem was provided by G. H. Hardy [5] employing the residue theorem and the well-known Mellin inversion theorem .