What is meant by a convergent sequence?

1 Answer
Vinicius M. G. Silveira
Jan 8, 2015

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_nan 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.

Related questions
See all questions in Limits of Infinite Sequences
Impact of this question
12275 views around the world
You can reuse this answer
Creative Commons License