Convergent Sequence

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

Formally, a sequence S_n converges to the limit S


if, for any epsilon>0, there exists an N such that |S_n-S|<epsilon for n>N. If S_n does not converge, it is said to diverge. This condition can also be written as


Every bounded monotonic sequence converges. Every unbounded sequence diverges.

See also

Conditional Convergence, Convergent, Limit, Strong Convergence, Weak Convergence

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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.

Referenced on Wolfram|Alpha

Convergent Sequence

Cite this as:

Weisstein, Eric W. "Convergent Sequence." From MathWorld--A Wolfram Web Resource.

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