Convergence Tests II: Alternating Series, Absolute Convergence, and the Ratio and Root Tests

Series with sign-alternating terms converge more readily than series of positive terms — the terms partially cancel. The Alternating Series Test formalises this and, crucially, provides an error bound: the truncation error is bounded by the first neglected term. This is the mechanism behind duration-convexity approximations: the Taylor series for bond price change $\Delta P \approx -D\cdot P\cdot\Delta y + \frac{1}{2}C\cdot P\cdot(\Delta y)^2 + \cdots$ is an alternating series in $\Delta y$, and the first-neglected term controls the approximation error. The Ratio Test is the main tool for power series and for any series involving factorials or $n$-th powers.

Learning objectives. By the end of this note you should be able to: