Limit value criterion

The limit criterion is a mathematical convergence criterion used to decide whether an infinite series is convergent or divergent.

statement

Let and be two infinite series with positive summands (that is, and for all ). Then applies ${\displaystyle \sum _{k}a_{k}}$ ${\displaystyle \sum _{k}b_{k}}$ ${\displaystyle a_{k}>0}$ ${\displaystyle b_{k}>0}$ ${\displaystyle k\in \mathbb {N} }$

• If the series is and converges , then it also converges . ${\displaystyle \limsup {\frac {a_{k}}{b_{k}}}<\infty }$ ${\displaystyle \sum b_{k}}$ ${\displaystyle \sum a_{k}}$
• If (which is equivalent to ), then the convergence of follows analogously from the convergence of . ${\displaystyle \liminf {\frac {a_{k}}{b_{k}}}>0}$ ${\displaystyle \limsup {\frac {b_{k}}{a_{k}}}<\infty }$ ${\displaystyle \sum a_{k}}$ ${\displaystyle \sum b_{k}}$
• If it applies at the same time , then and have the same convergence behavior. ${\displaystyle 0<\liminf {\frac {a_{k}}{b_{k}}}\leq \limsup {\frac {a_{k}}{b_{k}}}<\infty }$ ${\displaystyle \sum a_{k}}$ ${\displaystyle \sum b_{k}}$

In particular:

• If the sequence converges to a value with , then the series converges if and only if the series converges. ${\displaystyle {\frac {a_{k}}{b_{k}}}}$ ${\displaystyle c}$ ${\displaystyle 0<c<\infty }$ ${\displaystyle \sum a_{k}}$ ${\displaystyle \sum b_{k}}$

proof

Is , so is and therefore for a suitable and all sufficiently great . According to the majorant criterion , the convergence of the series results in the convergence of . ${\displaystyle \limsup {\frac {a_{k}}{b_{k}}}<\infty }$ ${\displaystyle {\frac {a_{k}}{b_{k}}}<C}$ ${\displaystyle a_{k}<C\,b_{k}}$ ${\displaystyle C}$ ${\displaystyle k}$ ${\displaystyle \sum b_{k}}$ ${\displaystyle \sum a_{k}}$

literature

• Harro Heuser : Textbook of Analysis . Part 1. Vieweg + Teubner, Wiesbaden 1980, ISBN 3-519-02221-4 (17th updated edition. Ibid. 2009, ISBN 978-3-8348-0777-9 ), pp. 204-205
• Rinaldo B. Schinazi: From Calculus to Analysis . Springer, 2011, ISBN 978-0-8176-8289-7 , p. 50
• Ed Barbeau: Fallacies, Flaws, and Flimflam . In: The College Mathematics Journal , Vol. 38, No. 2, March 2007, pp. 131-134, JSTOR 27646447
• Michele Longo, Vincenzo Valori: The Comparison Test: Not Just for Nonnegative Series . In: Mathematics Magazine , Vol. 79, No. 3, June 2006, pp. 205-210 ( JSTOR 27642937 )
• J. Marshall Ash: The Limit Comparison Test Needs Positivity . In: Mathematics Magazine , Vol. 85, No. 5, December 2012, pp. 374–375, doi: 10.4169 / math.mag.85.5.374

Web links

• Oswald Riemenschneider: Analysis II (PDF) script, Uni Hamburg, Theorem 16.33
• Eric W. Weisstein : limit comparison test . In: MathWorld (English).
• Paul's Online Notes on Comparison Test