What is meant by a convergent sequence?

1 Answer

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.