TOPICS
Search

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

lim_(n->infty)S_n=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

lim_(n->infty)^_S_n=lim_(n->infty)__S_n=S.

Every bounded monotonic sequence converges. Every unbounded sequence diverges.

See also

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

Explore with Wolfram|Alpha

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

Subject classifications