Answer to Question #142666 in Calculus for Moel Tariburu

Question

Determine whether or not the following series converges. If it converges, find its sum. ∑_(k=1)^∞▒1/7k

Answer 1

$\sum\limits_{k = 1}^\infty {\frac{1}{{7k}}}$

We apply the integral Cauchy criterion

$\int\limits_1^\infty {\frac{{dx}}{{7x}}} = \frac{1}{7}\mathop {\lim }\limits_{t \to \infty } \ln \left. x \right|_1^t = \frac{1}{7}\mathop {\lim }\limits_{t \to \infty } \left( {\ln t - \ln 1} \right) = \frac{1}{7}\mathop {\lim }\limits_{t \to \infty } \ln = \infty$

The series diverges since the corresponding improper integral diverges