alternating series test
The alternating series test^{}, or the Leibniz’s Theorem, states the following:
Theorem [1, 2] Let ${({a}_{n})}_{n=1}^{\mathrm{\infty}}$ be a non-negative, non-increasing sequence or real numbers such that ${lim}_{n\to \mathrm{\infty}}{a}_{n}=0$. Then the infinite series ${\sum}_{n=1}^{\mathrm{\infty}}{(-1)}^{(n+1)}{a}_{n}$ converges.
This test provides a necessary and sufficient condition for the convergence of an alternating series^{}, since if ${\sum}_{n=1}^{\mathrm{\infty}}{a}_{n}$ converges then ${a}_{n}\to 0$.
Example: The series ${\sum}_{k=1}^{\mathrm{\infty}}\frac{1}{k}$ does not converge, but the alternating series ${\sum}_{k=1}^{\mathrm{\infty}}{(-1)}^{k+1}\frac{1}{k}$ converges to $\mathrm{ln}(2)$.
References
- 1 W. Rudin, Principles of Mathematical Analysis, McGraw-Hill Inc., 1976.
- 2 E. Kreyszig, Advanced Engineering Mathematics, John Wiley & Sons, 1993, 7th ed.
Title | alternating series test |
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Canonical name | AlternatingSeriesTest |
Date of creation | 2013-03-22 12:27:09 |
Last modified on | 2013-03-22 12:27:09 |
Owner | Koro (127) |
Last modified by | Koro (127) |
Numerical id | 18 |
Author | Koro (127) |
Entry type | Theorem |
Classification | msc 40A05 |
Classification | msc 40-00 |
Synonym | Leibniz’s theorem |
Synonym | Leibniz test |
Related topic | AlternatingSeries |