Question

What is Ramanujan summation?

Answer 1

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.