I don't know of any use behind the calculation of the value for this particular series. But Ramanujan summation (and other summation methods for divergent serie

I don't know of any use behind the calculation of the value for this particular series. But Ramanujan summation (and other summation methods for divergent series) show up in the renormalization calculations in field theories. So this might sound like a random combination of fancy sounding words,but to give an idea of the connection: imagine you have some kind of integral that you want to evaluate or ODE you want to solve and you have some expansion that turns it into a series. But then you notice that a naive grouping of the terms in the series makes the sum go to infinity. But there are other ways where you can cleverly rearrange the grouping of terms so that it converges. So choosing a method for which this works (and gives a results that makes sense physically or agrees with other known exact results) is a serious business.