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Gilbert Strang, Calculus, 3rd edition (Wellesley-Cambridge Press; MIT OpenCourseWare RES.18-001) §10.2 Convergence Tests: Positive Series, #43 · Calculus · Series

(a) Explain why … is positive by using rectangles as in Figure…

(a) Explain why $D_{n} = (1 + \frac{1}{2} + \dots + \frac{1}{n}) - ln n$ is positive by using rectangles as in Figure 10.2.
(b) Show that $D_{n + 1}$ is less than $D_{n}$ by proving that
$\frac{1}{n + 1} < \int_{n}^{n + 1} \frac{d x}{x} .$
(c) (Calculator) The decreasing $D_{n}$ 's must approach a limit. Compute them until they go below $.6$ and $.58$ (when ?). The limit of the $D_{n}$ is Euler's constant $γ = .577 \dots$

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Gilbert Strang, Calculus, 3rd edition (Wellesley-Cambridge Press; MIT OpenCourseWare RES.18-001) · §10.2 Convergence Tests: Positive Series, #43

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