Ramanujan summation

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The Ramanujan Summation also has had a big impact in the area of general physics, specifically in the solution to the phenomenon know as the Casimir Effect.^[1]
Ramanujan summation is a technique invented by the mathematician Srinivasa Ramanujan for assigning a value to divergent infinite series.^[2]
Ramanujan summation essentially is a property of the partial sums, rather than a property of the entire sum, as that doesn't exist.^[2]
In the following text, ( ℜ ) {\displaystyle (\Re )} indicates "Ramanujan summation".^[2]
More advanced methods are required, such as zeta function regularization or Ramanujan summation.^[3]
Smoothing is a conceptual bridge between zeta function regularization, with its reliance on complex analysis, and Ramanujan summation, with its shortcut to the Euler–Maclaurin formula.^[3]
Ramanujan summation is a method to isolate the constant term in the Euler–Maclaurin formula for the partial sums of a series.^[3]
The regularity requirement prevents the use of Ramanujan summation upon spaced-out series like 0 + 2 + 0 + 4 + ⋯, because no regular function takes those values.^[3]