Answered: cos(n /3) n? n=1

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter7: Analytic Trigonometry

Section7.2: Trigonometric Equations

Problem 41E

See similar textbooks

Related questions

Question

Determine whether the series converges absolutely or conditionally, or diverges.

Expert Solution

Step 1

Given series,

$\sum_{n = 1}^{\infty} a_{n} = \sum_{n = 1}^{\infty} \frac{\cos (\frac{n}{3})}{n^{2}}$

Step 2

Using,

If $\sum_{n = 1}^{\infty} |a_{n}|$ is convergent then $\sum_{n = 1}^{\infty} a_{n}$ is absolutely convergent.

If $\sum_{n = 1}^{\infty} |a_{n}|$ is divergent then $\sum_{n = 1}^{\infty} a_{n}$ conditionally convergent.

Now

$\begin{array}{rcl} \sum_{n = 1}^{\infty} |a_{n}| & = & \sum_{n = 1}^{\infty} |\frac{\cos (\frac{n}{3})}{n^{2}}| \\ = & \sum_{n = 1}^{\infty} \frac{|\cos (\frac{n}{3})|}{|n^{2}|} using - 1 \leq \cos x \leq 1 s o, |\cos x| \leq 1 \\ = & \sum_{n = 1}^{\infty} \frac{1}{n^{2}} \end{array}$

$\sum_{n = 1}^{\infty} \frac{1}{n^{2}}$ is convergent (By the P-series test, convergent if p>1)

$\Rightarrow$ $\sum_{n = 1}^{\infty} \frac{\cos (\frac{n}{3})}{n^{2}}$ is absolutely convergent.

Step by step

Solved in 3 steps

Knowledge Booster

$Series$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.

Recommended textbooks for you

Algebra & Trigonometry with Analytic Geometry

Algebra

ISBN:

9781133382119

Author:

Swokowski

Publisher:

Cengage

Algebra & Trigonometry with Analytic Geometry

Algebra

ISBN:

9781133382119

Author:

Swokowski

Publisher:

Cengage

SEE MORE TEXTBOOKS