Alternating Series Test (Leibniz Test)

Alternating Series Test: Let $(b_n)$ be a decreasing sequence of nonnegative real numbers with $b_n\to 0$. Then the alternating series $\sum_{n=1}^\infty (-1)^{n-1} b_n$ converges .

This test explains convergence driven by cancellation even when $\sum b_n$ diverges, and it yields a standard remainder estimate: the truncation error is at most the next term magnitude.

Proof sketch (optional): The even and odd partial sums form monotone sequences that are bounded and squeeze to the same limit, using the monotonicity of $b_n$.