What is Ramanujan summation?

1 Answer
Teddy
Jun 15, 2018

Ramanujan summation is a way to assign a finite value to a divergent series.

Explanation:

Ramanujan summation allows you to manipulate sums without worrying about operations on infinity that would be considered wrong. For example, you can use Ramanujan summation to assign a finite value to the infinite series #1-1+1-1+1-...#, which we know diverges.

Let #S=1-1+1-1+1-...#
#1-S=1-(1-1+1-1+1-...)#
#1-S=1-1+1-1+1-...#
#1-S=S#
Therefore, #S=1/2# which means Ramanujan summation assigns the value of #1/2# to the divergent series #1-1+1-1+1-...# even though we know the series doesn't approach a single value. A famous result of Ramanujan summation is #1+2+3+4+5+...=-1/(12)#, which is used in string theory.

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