# alternating series test

The , or the Leibniz’s Theorem, states the following:

Theorem [1, 2] Let $(a_{n})_{n=1}^{\infty}$ be a non-negative, non-increasing sequence or 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.
Title alternating series test AlternatingSeriesTest 2013-03-22 12:27:09 2013-03-22 12:27:09 Koro (127) Koro (127) 18 Koro (127) Theorem msc 40A05 msc 40-00 Leibniz’s theorem Leibniz test AlternatingSeries