Can someone let me know the difference between a convergent series and an asymptotic series with an example? Can both the series be the same at some situations? In what situations an asymptotic series is more useful?
2026-03-27 13:27:30.1774618050
Difference between a convergent series and an asymptotic series?
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A convergent series gets closer to the sum as more terms are taken.
A asymptotic series initially gets closer to the sum but, after a while, gets increasingly further away and usually ultimately diverges. In many asymptotic series, the error is less than the last term. Since the terms eventually get large, the usual practice is to stop the series when the terms no longer decrease
A classic example of an asymptotic series is Stirling's formula. One of many discussions is here: http://en.wikipedia.org/wiki/Stirling%27s_approximation