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Convergence | Definition, Examples, & Facts | Britannica Ask the Chatbot Games & Quizzes History & Society Science & Tech Biographies Animals & Nature Geography & Travel Arts & Culture ProCon Money Videos convergence Introduction References & Edit History Related Topics Quizzes Numbers and Mathematics Contents CITE verified Cite While every effort has been made to follow citation style rules, there may be some discrepancies. Please refer to the appropriate style manual or other sources if you have any questions. Select Citation Style MLA APA Chicago Manual of Style Copy Citation Share Share Share to social media Facebook X URL https://www.britannica.com/science/convergence-mathematics Feedback External Websites Feedback Corrections? Updates? Omissions? Let us know if you have suggestions to improve this article (requires login). Feedback Type Select a type (Required) Factual Correction Spelling/Grammar Correction Link Correction Additional Information Other Your Feedback Submit Feedback Thank you for your feedback Our editors will review what you’ve submitted and determine whether to revise the article. External Websites Wolfram MathWorld - Convergent Series California State University San Marcos - Math Lab - Testing for Convergence or Divergence of a Series convergence mathematics Ask Anything Homework Help Written and fact-checked by Britannica Editors Encyclopaedia Britannica's editors oversee subject areas in which they have extensive knowledge, whether from years of experience gained by working on that content or via study for an advanced degree.... Britannica Editors History Britannica AI Ask Anything Table of Contents Table of Contents Ask Anything convergence , in mathematics , property (exhibited by certain infinite series and functions ) of approaching a limit more and more closely as an argument (variable) of the function increases or decreases or as the number of terms of the series increases. For example, the function y = 1/ x converges to zero as x increases. Although no finite value of x will cause the value of y to actually become zero, the limiting value of y is zero because y can be made as small as desired by choosing x large enough. The line y = 0 (the x -axis) is called an asymptote of the function. Key People: James Gregory (Show more) Related Topics: uniform convergence Weierstrass M-test sequence (Show more) See all related content Similarly, for any value of x between (but not including) −1 and +1, the series 1 + x + x 2 +⋯+ x n converges toward the limit 1/(1 − x ) as n , the number of terms, increases. The interval −1 < x < 1 is called the range of convergence of the series; for values of x outside this range, the series is said to diverge. Britannica Quiz Numbers and Mathematics This article was most recently revised and updated by Adam Augustyn .