Convergent Sequence

A sequence is said to be convergent if it approaches some limit (D'Angelo and West 2000, p. 259).

Formally, a sequence  converges to the limit 

if, for any , there exists an  such that  for . If  does not converge, it is said to diverge. This condition can also be written as

Every bounded monotonic sequence converges. Every unbounded sequence diverges.

REFERENCES:

D'Angelo, J. P. and West, D. B. Mathematical Thinking: Problem-Solving and Proofs, 2nd ed. Upper Saddle River, NJ: Prentice-Hall, 2000.

Jeffreys, H. and Jeffreys, B. S. "Bounded, Unbounded, Convergent, Oscillatory." §1.041 in Methods of Mathematical Physics, 3rd ed. Cambridge, England: Cambridge University Press, pp. 11-12, 1988.