| It seems like you're getting sequences confused with series. To show that a sequence (which in simplest terms is just a list of numbers) converges, you do take the limit as n-> infinity. If that limit exists and is finite, the sequence converges. Since it is well known that sin(x)/x -> 1 as x -> inf, the sequence {sin(n^(1/2)) / n^(1/2) } converges.
However, if you are dealing with a series, which is when you sum a list of terms, there are many different methods for showing convergence/divergence. The one that is encountered early on is known as the test for divergence. Here, you take the limit of the general term as n -> inf and if that limit does not exist or does not equal 0, then the series diverges. Note that this does not tell us whether the series converges, however if it is conclusive, then we know the series diverges. |