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Gilbert Strang, Calculus, 3rd edition (Wellesley-Cambridge Press; MIT OpenCourseWare RES.18-001) §10.2 Convergence Tests: Positive Series, #45 · Calculus · Series

(a) Show that … by adding … to both sides

(a) Show that $1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} - \dots - \frac{1}{2 n} = \frac{1}{2 n + 1} + \dots + \frac{1}{2 n}$ by adding $2 (\frac{1}{2} + \frac{1}{4} + \dots + \frac{1}{2 n})$ to both sides.
(b) Why is the right side close to $ln 2 n - ln n$ ? Deduce that $1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \dots$ approaches $ln 2$ .

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Gilbert Strang, Calculus, 3rd edition (Wellesley-Cambridge Press; MIT OpenCourseWare RES.18-001) · §10.2 Convergence Tests: Positive Series, #45

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