Answered: 1 sinh k k=1

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter5: Inverse, Exponential, And Logarithmic Functions

Section: Chapter Questions

Problem 44RE

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Question

Use a convergence test of your choice to determine whether the following series converge.

Expert Solution

Step 1 Given

$\sum_{k = 1}^{\infty} \frac{1}{\sin h k}$

Now check given series converges or not

So here use ratio test to determine whether the following series converges.

Ratio test

1) If $\lim_{n \to \infty} |\frac{a_{n + 1}}{a_{n}}| = L < 1$ then the series $\sum_{n = 1}^{\infty} a_{n}$ is absolutely convergent.

2)If $\lim_{n \to \infty} |\frac{a_{n + 1}}{a_{n}}| = L > 1$ or $\lim_{n \to \infty} |\frac{a_{n + 1}}{a_{n}}| = \infty$ then the series $\sum_{n = 1}^{\infty} a_{n}$ is divergent.

3) If $\lim_{n \to \infty} |\frac{a_{n + 1}}{a_{n}}| = 1$ then ratio test is inconclusive that is no conclusion.

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