## 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 to neither or 0 depending on the values of r and a:  1, -3, 9, -27... The terms of this series alternate because The 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 let then 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. If then therefore the sequence has a limit. Since L is positive  