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Gilbert Strang, Calculus, 3rd edition (Wellesley-Cambridge Press; MIT OpenCourseWare RES.18-001) §10.2 Convergence Tests: Positive Series, #50 · Calculus · Series

Construct a series … that converges faster than … but slower than…

The $n$ th prime number $p_{n}$ satisfies $p_{n} / n ln n \to 1$ . Prove that
$\sum \frac{1}{p _{n}} = \frac{1}{2} + \frac{1}{3} + \frac{1}{5} + \frac{1}{7} + \frac{1}{11} + \dots diverges.$

Construct a series $Σ a_{n}$ that converges faster than $Σ b_{n}$ but slower than $Σ c_{n}$ (meaning $a_{n} / b_{n} \to 0$ , $a_{n} / c_{n} \to \infty$ ).

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Gilbert Strang, Calculus, 3rd edition (Wellesley-Cambridge Press; MIT OpenCourseWare RES.18-001) · §10.2 Convergence Tests: Positive Series, #50

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