Sequences and Limits

Sequences can be defined in various ways, in iterative and closed form. We have so far analysed geometric and arithmetic series. The terms of an arithmetic series always tend to(we say the series does not tend to a limit):

The terms of a geometric series may tend toneither or 0 depending on the values of r and a:

1, -3, 9, -27... The terms of this series alternate becauseThe series does not tend to a limit or

This series tends to the limit 0, since

There are series for which a limit exists which is non zero. For example:

u-0=1

(1)

Carrying on in this way we obtain the sequence 1.684, 1.763, 1.822, 1.867, 1.900, 1.925, 1.944, 1.958, 1.968, 1.976, 1.972, 1.987, 1.990...

The terms are getting closer together. We may call the limit L, and to find L we letthen We can substitute into (1) to obtain

Example:

Show that the sequence tends to a limit L and find L.

All the terms of the sequence are positive and the terms of the sequence are increasing. Ifthentherefore the sequence has a limit.

Since L is positive