Question

How do I find the limit of a sequence?

Answer 1

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,

`{1/n}_n = {1, 1/2, 1/3,....}`

In this case the limit turns out to be `0` because the higher the terms are, the closer they get to `0` and for a sufficiently large `n` which is infinity, `Lim 1/n = 0` which is the limit point.

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, `0` does not represent any term of the `{1/n}_n` sequence.

There can be other types of sequences such as the ones in which the consecutive terms increase in magnitude for higher values of `n`. In this case, difference between two consecutive terms increases and the sequence diverges altogether. A good example is,

`{n}_n = {1, 2, 3, ....}`

Here, `Lim n = prop`

There can be another type of sequence known as the oscillating sequence as shown,

`{(-1)^n}_n = {-1,1,-1,1,....}`

The series neither converges not diverges and is termed as oscillatory.