How do I find the limit of a sequence?
There is no general way of determining the limit of a sequence. Also, not all sequences have limits. However, if a sequence has a limit point, it must be unique. (This is an elementary result of analysis).
If a sequence be such that, the higher and higher terms get smaller in magnitude or alternatively the difference between consecutive terms decreases as the order of terms increase, you can have a limit. A pretty good example would be,
In this case the limit turns out to be
Two things may be observed. First the limit point has to be unique for every sequence. Second, the limit point may not be a member of the sequence itself as in this case,
There can be other types of sequences such as the ones in which the consecutive terms increase in magnitude for higher values of
There can be another type of sequence known as the oscillating sequence as shown,
The series neither converges not diverges and is termed as oscillatory.