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# 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.

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

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## 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.

## Referenced on Wolfram|Alpha

Convergent Sequence

## Cite this as:

Weisstein, Eric W. "Convergent Sequence." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ConvergentSequence.html