Sum Of All Natural Number Ramanujan Summation 1 2 ођ The partial sums of the series 1 2 3 4 5 6 ⋯ are 1, 3, 6, 10, 15, etc.the nth partial sum is given by a simple formula: = = ( ). this equation was known. The great debate over whether 1 2 3 4 ∞ = 1 12. can the sum of all positive integers = 1 12? mathematician srinivasa ramanujan, of an infinite number of terms of the series: 1 2.
Sum Of All Natural Numbers Ramanujan Infinite Sum 1 2ођ Ramanujan summation is a technique invented by the mathematician srinivasa ramanujan for assigning a value to divergent infinite series.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. So, 2s (2) is 1 2 such that the value of s (2) is ¼. next, subtract s (2) from s, which gives: 1 2 3 4 … – (1 2 3 4 …) = 0 4 0 8 0 12 0 16…. which can be also be written as 4 times (1 2 3 4…) or 4s. now, we’ve got the hat, we just need to point and cast the spell. we have shown that s s (2) = 4s, but s (2) is equal to ¼. Recently a very strange result has been making the rounds. it says that when you add up all the natural numbers 1 2 3 4 then the answer to this sum is 1 12. the idea featured in a numberphile video (see below), which claims to prove the result and also says that it's used all over the place in physics. \(1 2 3 4 \cdots = (1 1 1 1 \cdots)^{2} = \large \frac{1}{(1 1)^{2}} \normalsize =\large \frac{1}{4}\) after getting all the important results let’s come back to ramanujan’s sum and his result. now you can easily understand how he wrote this:.
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