Question

What is meant by a convergent sequence?

Answer 1

A sequence is said to be convergent if it's limit exists.

Else, it's said to be divergent.

It must be emphasized that if the limit of a sequence `a_n` is infinite, that is `lim_(n to oo) a_n = oo` or `lim_(n to oo) a_n = -oo`, the sequence is also said to be divergent.

A few examples of convergent sequences are:

• `1/n`, with `lim_(n to oo) 1/n = 0`
• The constant sequence `c`, with `c in RR` and `lim_(n to oo) c = c`
• `(1+1/n)^n`, with `lim_(n to oo) (1+1/n)^n = e` where `e` is the base of the natural logarithms (also called Euler's number).

Convergent sequences play a very big role in various fields of Mathematics, from estabilishing the foundations of calculus, to solving problems in Functional Analysis, to motivating the development of Toplogy.