Does the sequence converge?
A sequence converges if there is a limit for its terms as the number terms goes to infinity.
Is there a limit of as goes to infinity? Let’s see:
. Since the limit as goes to infinity of equals 0, the limit as goes to infinity of .
There is a limit: . Therefore the sequence converges.
There may be confusion between convergence of a sequence and conversion of a series. Note the distinction: A sequence is a list of numbers: one number, then another number, then another, etc.; while a series is a sum of numbers: one number plus another number, plus another, etc. More confusing, there is a whole topic about convergence of series, with several kinds of tests for convergence.
To determine whether a sequence converges, don’t use the convergence tests that you use for series; instead, look at the limit. While a series converges only if the sum of the terms is limited, a sequence converges if its th term is limited as goes to infinity.
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More information about sequences: