Answered: k sin?k 1 + k3

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k sin?k 1 + k3

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter7: Analytic Trigonometry

Section7.4: Multiple-angle Formulas

Problem 26E

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Question

100%

Determine whether the series is convergent or divergent.

Expert Solution

Step 1

Let,

$a_{k} = \frac{k \sin^{2} (k)}{1 + k^{3}}$

Step 2

$\begin{array}{rcl} \lim_{k \to \infty} \frac{k \sin^{2} (k)}{1 + k^{3}} [\equiv \frac{\infty}{\infty}] \\ = & \lim_{k \to \infty} \frac{\sin^{2} (k) + 2 k \sin (k) \cos (k)}{3 k^{2}} [\equiv \frac{\infty}{\infty}] \\ = & \lim_{k \to \infty} \frac{2 \sin (2 k) + 2 k \cos (k)}{6 k} [\equiv \frac{\infty}{\infty}] \\ = & \lim_{k \to \infty} \frac{6 \cos (2 k) - 4 \sin (2 k)}{6} [\equiv \frac{\infty}{\infty}] \\ = & \infty \end{array}$

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