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Gilbert Strang, Calculus, 3rd edition (Wellesley-Cambridge Press; MIT OpenCourseWare RES.18-001) §10.3 Convergence Tests: All Series, #42 · Calculus · Series

### *43 Show that the tangent of … is

The sums in Problem 41 form a Cantor set centered at zero. What is the smallest positive number in the set? Choose signs to show that $\frac{1}{4}$ is in the set.

### *43
Show that the tangent of $θ = \frac{1}{2} (π - 1)$ is $sin 1/ (1 - cos 1)$ . This is the imaginary part of $s = - ln (1 - e^{i})$ . From $s = Σ e^{in} / n$ deduce the remarkable sum $Σ (sin n) / n = \frac{1}{2} (π - 1)$ .

This exercise builds on another exercise in its source book, which isn't shown here.

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Gilbert Strang, Calculus, 3rd edition (Wellesley-Cambridge Press; MIT OpenCourseWare RES.18-001) · §10.3 Convergence Tests: All Series, #42

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