A Quick Note on Sequences

A sequence is usually thought of as a list of numbers, such as the sequence
1, 4, 9, 16, 25, ...

and we often want to know whether the sequence converges or diverges, and if it converges, what its limit value is.

All of the discussion about limits on these pages will refer to functions of real variables. However, these results can be fully adapted to sequences, due to the following result:

If the limit as x goes to infinity
of f(x) = L and f(x) = a[n] when n is an integer, then

the limit as n goes to infinity
of a[n] = L.

From this, we see that all of the limit laws, and l'Hôpital's Rule, hold for sequences as well as functions of real variables.

Copyright © 1996 Department of Mathematics, Oregon State University

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