It just occurred to me that I tend to think of integrals primarily as indefinite integrals and sums primarily as definite sums. That is, when I see a definite integral, my first approach at solving it is to find an antiderivative, and only if that doesn't seem promising I'll consider whether there might be a special way to solve it for these special limits; whereas when I see a sum it's usually not the first thing that occurs to me that the terms might be differences of some function. In other words, telescoping a sum seems to be just one particular way among many of evaluating it, whereas finding antiderivatives is the primary way of evaluating integrals. In fact I learned about telescoping sums much later than about antiderivatives, and I've only relatively recently learned to see these two phenomena as different versions of the same thing. Also it seems to me that empirically the fraction of cases in which this approach is useful is much higher for integrals than for sums.
So I'm wondering why that is. Do you see a systematic reason why this method is more productive for integrals? Or is it perhaps just a matter of education and an "objective" view wouldn't make a distinction between sums and integrals in this regard?
I'm aware that this is rather a soft question, but I'm hoping it might generate some insight without leading to open-ended discussions.
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