The alternating series test, 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 $\sum_{n=1}^\infty b_n$ 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 $\ln(2)$ .

Bibliography

1

W. Rudin, Principles of Mathematical Analysis, McGraw-Hill Inc., 1976.

2

E. Kreyszig, Advanced Engineering Mathematics, John Wiley & Sons, 1993, 7th ed.