In another thread the prospect of infinite sums came up and I wanted to introduce one of the oddest results I've ever seen in math. Its a situation where clearly the partial sums tend towards infinite, and where its possible to develops means of summing up the infinite numbers in such a way that it returns a specific result.

Let's dive right into it and give the result.

This ridiculously counterintuitive result stems from a field called Ramanujan summanation, which is one in a group of summation methods for infinite series that are divergent. Strangely enough some of these methods can be used to sum up the infinite series of integers, and they always end up given the result above.

I'll show you how this result can be derived from something called Abel summation.

Now if we take another series, namely

Then naturally we'd try to solve this by finding its partial sum (summing up all but the leading term) and seeing what this converges to

Obviously the partial sum is divergent as we add more and more terms. So we'll need a different way of summing. The mathematician Abel proposed (though Euler found it first), the following means of handling this type of summation.

This can't sum the first series mentioned, but it is able to sum the second one, giving us...

From this we can determine what series mentioned in the beginning will be, if it has any value at all. This is done simple be substracting the partial sum of one series from the other.

Of course this result can also be found directly, but that requires a stronger method of summation than Abel summation, such as Ramanujan summation. However unsurprisingly it yields the exact same answer. And its not merely a theoretical answer, as sums over all the natural integers occur often in quantum field theory, and in the derivation of the Casimir effect one has to use such a sum.

Therefore this result, along with various strong derivations are included in Advanced Quantum Mechanics course work.

What do you guys think?

Let's dive right into it and give the result.

This ridiculously counterintuitive result stems from a field called Ramanujan summanation, which is one in a group of summation methods for infinite series that are divergent. Strangely enough some of these methods can be used to sum up the infinite series of integers, and they always end up given the result above.

I'll show you how this result can be derived from something called Abel summation.

Now if we take another series, namely

Then naturally we'd try to solve this by finding its partial sum (summing up all but the leading term) and seeing what this converges to

Obviously the partial sum is divergent as we add more and more terms. So we'll need a different way of summing. The mathematician Abel proposed (though Euler found it first), the following means of handling this type of summation.

This can't sum the first series mentioned, but it is able to sum the second one, giving us...

From this we can determine what series mentioned in the beginning will be, if it has any value at all. This is done simple be substracting the partial sum of one series from the other.

Of course this result can also be found directly, but that requires a stronger method of summation than Abel summation, such as Ramanujan summation. However unsurprisingly it yields the exact same answer. And its not merely a theoretical answer, as sums over all the natural integers occur often in quantum field theory, and in the derivation of the Casimir effect one has to use such a sum.

Therefore this result, along with various strong derivations are included in Advanced Quantum Mechanics course work.

What do you guys think?

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