Ramanujan summation is a technique invented by the mathematician Srinivasa Ramanujan for assigning a value to infinite divergent series.
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Count the number of ordered triples of positive integers whose product is not greater than a given number?
Given N, count the number of ordered triplets(a,b,c) whose product $abc \leq N$. I have found the series here . But I am not sure I understand Benoit Cloitre work which proposed an efficent way to ... riemann-hypothesis divisor-counting-function ramanujan-summation- 163
Grandi's series
So a couple of days ago I decided to learn why Ramanujan's theory 1+2+3+4+... = -1/12 is the way it is. The first step in the proof/derivation was to consider Grandi's series A = (1-1+1-1+1-1+1-1...) ... ramanujan-summation- 21
How $\phi(2)$ comes into the picture?
We know Ramanujan's $$\phi(a,n) = 1+ 2\sum_{k =0}^{n}{\frac{1}{(ak)^3-ak}}$$ How can I prove $$\sum_{k=1}^n\frac{1}{n+k} = \frac{n}{2n+1} + \sum_{k=1}^n(\frac{1}{(2k)^3-2k})$$ Don't know how to prove ... calculus sequences-and-series analysis logarithms ramanujan-summation- 3
Why/How was Hardy impressed with Ramanujan’s work prior to seeing any proofs of their validity?
Hardy’s famous correspondence with Ramanujan highlights an interesting aspect of the mathematics discipline; the appreciation of a mathematical expression prior to knowing its validity. Why/how was ... soft-question continued-fractions ramanujan-summation- 256
A dig at Ramanujan's: $\sum_{k=1}^{\infty} (-1)^{k-1} \frac{x^{pk}}{k(k!)^p} \sim p \ln x +p \gamma,~ p>0$
Ramanujan's claim on page 98, in the book (`Ramanujan's note book part 1' by Bruce C. Berndt) is that $$\sum_{k=1}^{\infty} (-1)^{k-1} \frac{x^{pk}}{k(k!)^p} \sim p \ln x +p \gamma,\qquad p>0\tag{1}... sequences-and-series convergence-divergence ramanujan-summation- 38k
Ramanujan summation of alternating factorials: why is it $0.596...$?
I was reading Ramanujan's letter to Hardy and came across this identity: $$0!-1!+2!-3!+...=0.596...$$ Ramanujan didn't give the exact value of that constant, but it is the Euler-Gompertz constant, as ... summation factorial ramanujan-summation- 401
Interpret "the sum of all natural numbers equal $-1/12$" without complex analysis skills
is it possible to show if there is a link between the sum of all natural numbers and the value $ - \frac {1} {12} $, through simple real analysis intuitions? In some contexts of physics this result ... convergence-divergence riemann-zeta ramanujan-summation- 1,507
Computing Ramanujan asymptotic formula from Rademacher's formula for the partition function
I am trying to derive the Hardy-Ramanujan asymptotic formula $$p(n) \sim \frac{1}{4n\sqrt{3}}e^{\pi\sqrt{\frac{2n}{3}}}$$ from Radmacher's formula for the partition function $p(n)$ given by $$p(n)=\... combinatorics number-theory elementary-number-theory integer-partitions ramanujan-summation- 107
What is the Ramanujan summation for the series $\sqrt[n]{2}$
A Ramanujan summation is a technique invented by the mathematician Srinivasa Ramanujan for assigning a value to divergent infinite series In my case, I'm interested in assigning a value to the ... limits divergent-series ramanujan-summation- 3,756
Proving $\int_{0}^{1} \frac{\tanh^{-1}\sqrt{x(1-x)}}{\sqrt{x(1-x)}}dx=\frac{1}{3}(8C-\pi\ln(2+\sqrt{3}))$ for an identity of Srinivasa Ramanujan
Ramanujan is supposed to have given more than five thousand elegant results, a good number of them are yet to be proved or disproved. Yesterday in the comment section of Proving that $ \sum_{k=0}^\... integration definite-integrals binomial-coefficients binomial-theorem ramanujan-summation- 38k
Find $\sum_{n=1}^{\infty} \frac{n^{\sigma -1} (n+\sigma )-(n+1)^{\sigma }}{\sigma(1-\sigma)}$ for $ 0<\sigma<1$
Find $\sum_{n=1}^{\infty} \frac{n^{\sigma -1} (n+\sigma )-(n+1)^{\sigma }}{\sigma(1-\sigma)}$ for $ 0<\sigma<1$ My try $ \sum_{n=1}^{\infty} \frac{n^{\sigma -1} (n+\sigma )-(n+1)^{\sigma }}{\... summation analytic-number-theory riemann-zeta riemann-hypothesis ramanujan-summationFound this identity by accident
I found the following identity when trying to prove something else: $$\sum\limits_{n=1}^\infty\frac{(n-1)!x}{\prod\limits_{k=1}^n (k+x)}\equiv 1,~ \forall x>0$$ I'm sure there's a name for it (or ... sequences-and-series ramanujan-summation- 131
Can the ramanujan summation formula be used to evaluate convergent series?
I know it's used to give values to divergent series, but when applied to absolutely convergent series, does it give the value at which the series converges? By the way since someone asked about how I ... sequences-and-series divergent-series ramanujan-summation- 129
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. Since ... group-theory ramanujan-summation- 27
What is Landau Ramanujan constant?
What is Landau Ramanujan constant? I searched on Wikipedia but explanation was not clear enough please explain it along with example thank you calculus number-theory prime-numbers mathematicians ramanujan-summation- 9
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