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one day in one of my high school math classes i was introduced to a sequence that im sure every mathematician has pondered at least once the sequence goes one then one plus two then one plus two plus three and so on the nth number of the sequence is the sum of the first n natural numbers the purpose of the lesson was to learn sigma notation how we can unambiguously express the sequence like this but the teacher threw in a bonus a clever proof that the sum is actually equal to n times n plus one divided by two this really fascinated me not just because of its elegance and simplicity but because it extends the domain of the sum the original sum is discrete it only makes sense to plug whole numbers into it for instance if we tried to plug in one half we wouldnt even know what to do but the new formula is continuous it has no problem handling any number you throw at it so if youre like me youll wonder what other sums can be extended this way well there are endless sums you could try m