**Determine whether each of the following sequences is convergent. Justify.****(a)** $(x_n)$ where $x_n = \frac{\sin(n\pi/6)}{n}$**(b)** $\left(\frac{c^{n+2} - d^{n+2}}{c^n + d^n}\right)$ where $0 < c < d$

(a) The given sequence is . Since is a bounded sequence and .Then Hence the sequence is…

Answered: Determine whether each of the following…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

See similar textbooks

Related questions

Question

**Determine whether each of the following sequences is convergent. Justify.**

**(a)** $(x_n)$ where $x_n = \frac{\sin(n\pi/6)}{n}$

**(b)** $\left(\frac{c^{n+2} - d^{n+2}}{c^n + d^n}\right)$ where $0 < c < d$

Expert Solution

Step 1: Given Sequence:

(a) The given sequence is $open parentheses x subscript n close parentheses space w h e r e space x subscript n equals fraction numerator sin open parentheses fraction numerator n pi over denominator 6 end fraction close parentheses over denominator n end fraction$ .

Since $open parentheses sin fraction numerator n pi over denominator 6 end fraction close parentheses$ is a bounded sequence and $limit as n rightwards arrow infinity of 1 over n equals 0$ .

Then $limit as n rightwards arrow infinity of fraction numerator sin open parentheses fraction numerator n pi over denominator 6 end fraction close parentheses over denominator n end fraction equals 0$

Hence the sequence $open parentheses x subscript n close parentheses$ is convergent.