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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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This is is interesting -- there's kind of a symmetry here: That the usual definition of an infinite sum is to find a sequence whose difference is the terms of the sum (i.e. the sequence of partial sums!) and then look for its limit -- but infinite sums are not usually computed that way. On the other hand, the official definition of a Riemann integral is as a (limiting) sum, but it is not usually computed that way. –  Henning Makholm Oct 14 '12 at 13:22
    
Perhaps there's a cognitive bias at work: sums that can easily be solvable by telescoping don't feel like we're really doing work; therefore we tend to underestimate the number of times we use that techniques. Similarly, when we can use geometric arguments to short-cut our way to a definite integal, it doesn't feel as if "evaluating integrals" is really what we're doing. –  Henning Makholm Oct 14 '12 at 13:27
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@Henning: Skew-symmetry, maybe? :-) –  Brian M. Scott Oct 14 '12 at 13:43

3 Answers 3

Your answer is probably buried within this statement: "I learned about telescoping sums much later than about antiderivatives."

All mathematicians, and a substantial fraction of college graduates, study integration for at least a year, frequently many more. Rather fewer study discrete techniques, and only specialists study difference calculus for a year or more. The analog to finding an antiderivative is finding a WZ pair, which is viewed as an "advanced technique".

Perhaps if we were to reverse the curriculum, teaching everybody differences and reserving differential calculus for specialists, your question might have been reversed. Incidentally, I think this might not be such a bad idea; continuous methods, which used to rule the world of science and engineering, are rapidly being replaced by discrete approximation and simulation.

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Also, read about how Feynman learned some non-standard methods of indefinite integration (such as differentiating under the integral sign) and used these to get various integrals that usually needed complex integration.

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I think telescoping sums is not the analogous concept to antidifferentiation for solving definite sums/integrals, because it's too specialized. When you find an antiderivative (for the purpose of evaluating a definite integral), you find a function corresponding to a definite integral from a fixed point: $x^2/2=\int_0^xt\,\mathrm dt$ helps you evaluate $\int_{12}^{517}t\,\mathrm dt$. For certain sums, you do the same: $n(n+1)/2=\sum_{i=1}^ni$ can help you evaluate $\sum_{i=12}^{517}i$. Admittedly, when you have a sum that telescopes, then you have a tidy formula for a sum from a fixed index (usually $1$) to an arbitrary one, but not every formula like that comes from a telescope.

I think the simplest reason it's more productive in the case of integrals is that there are more elementary antiderivatives than "definite sum formulae". A sum of unit fractions can be written as a difference in harmonic numbers, but because harmonic numbers don't have a tidy elementary formula (like $\log (1+x)$), this isn't too helpful.

Also, while summation by parts is useful, arbitrary substitutions are not as easy as they are with integrals: To get the formula for the sum of the squares of the first $n$ odd numbers can be done with the sum of the first $n$ squares, but it's not quite as easy as substitution for calculating antiderivatives, and there is no general method for things like this.

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