alternating series test

The alternating series test, orthe Leibniz’s Theorem, states the following:

Theorem [1, 2]Let $(a_{n})_{n=1}^{\\infty}$ be a non-negative, non-increasing sequenceor real numbers such that $\\lim_{n\\rightarrow\\infty}a_{n}=0$ .Then the infinite series $\\sum_{n=1}^{\\infty}(-1)^{(n+1)}a_{n}$ converges.

This test provides a necessary and sufficient condition for the convergence of an alternating series, since if $\\sum_{n=1}^{\\infty}a_{n}$ converges then $a_{n}\\to 0$ .

Example: The series $\\sum_{k=1}^{\\infty}\\frac{1}{k}$ does not converge, but the alternating series $\\sum_{k=1}^{\\infty}(-1)^{k+1}\\frac{1}{k}$ converges to $\\ln(2)$ .

References

1 W. Rudin, Principles of Mathematical Analysis, McGraw-Hill Inc., 1976.
2 E. Kreyszig,Advanced Engineering Mathematics,John Wiley & Sons, 1993, 7th ed.

单词	AlternatingSeriesTest
释义	alternating series test The alternating series test, orthe Leibniz’s Theorem, states the following: Theorem [1, 2]Let $(a_{n})_{n=1}^{\\infty}$ be a non-negative, non-increasing sequenceor real numbers such that $\\lim_{n\\rightarrow\\infty}a_{n}=0$ .Then the infinite series $\\sum_{n=1}^{\\infty}(-1)^{(n+1)}a_{n}$ converges. This test provides a necessary and sufficient condition for the convergence of an alternating series, since if $\\sum_{n=1}^{\\infty}a_{n}$ converges then $a_{n}\\to 0$ . Example: The series $\\sum_{k=1}^{\\infty}\\frac{1}{k}$ does not converge, but the alternating series $\\sum_{k=1}^{\\infty}(-1)^{k+1}\\frac{1}{k}$ converges to $\\ln(2)$ . References 1 W. Rudin, Principles of Mathematical Analysis, McGraw-Hill Inc., 1976. 2 E. Kreyszig,Advanced Engineering Mathematics,John Wiley & Sons, 1993, 7th ed.
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