Exact Answers are great when we can find them; there’s something very satisfying about complete knowledge. But there’s also a time when approximations are in order. If we run into a sum or a recurrence whose solution doesn’t have a closed form (as far as we can tell), we still would like to know something about the answer; we don’t have to insist on all or nothing. And even if we do have a closed form, our knowledge might be imperfect, since we might not know how to compare it with other closed forms.
Uh oh . . . here comes that A-word.
For example, there is (apparently) no closed form for the sum
But it is nice to know that
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