Answered: 1 1 n+ n=1

Algebra: Structure And Method, Book 1

(REV)00th Edition

ISBN:9780395977224

Author:Richard G. Brown, Mary P. Dolciani, Robert H. Sorgenfrey, William L. Cole

Publisher:Richard G. Brown, Mary P. Dolciani, Robert H. Sorgenfrey, William L. Cole

Chapter4: Polynomials

Section4.2: Adding And Subtracting Polynomials

Problem 19OE

See similar textbooks

Related questions

Question

In Exercises 17-32, determine convergence or divergence by any method.

Expert Solution

Step 1

Given:

$\sum_{n = 1}^{\infty} \frac{1}{n + \frac{1}{n}}$

To Determine:

Whether the given series is convergent or divergent.

Concept:

$(i) . p - series Test: The p - series is given by: \sum_{n = 1}^{\infty} \frac{1}{n^{p}} = \frac{1}{1^{p}} + \frac{1}{2^{p}} + \frac{1}{3^{p}} + \cdot \cdot \cdot \cdot \cdot, where p > 0 \to If p > 1, then the series converges. \to if 0 < p \leq 1, then the series diverges. \Rightarrow \sum_{n = 1}^{\infty} \frac{1}{n} is divergent. (i i) . Limit Comparison Test: Let \sum_{n = 1}^{\infty} a_{n} and \sum_{n = 1}^{\infty} b_{n} are two series, such that L = \lim_{n \to \infty} \frac{a_{n}}{b_{n}} (i) . If L is finite and positive, then either both series converges or diverges. (i i) . If L = 0, and if \sum_{n = 1}^{\infty} b_{n} converges then \sum_{n = 1}^{\infty} a_{n} converges as well. (i i i) . If L = \infty, and if \sum_{n = 1}^{\infty} b_{n} diverges then \sum_{n = 1}^{\infty} a_{n} diverges as well.$

Step 2

Explanation:

$As given: \sum_{n = 1}^{\infty} \frac{1}{n + \frac{1}{n}} = \sum_{n = 1}^{\infty} \frac{n}{n^{2} + 1} let a_{n} = \frac{n}{n^{2} + 1} and b_{n} = \frac{1}{n} ∴ \lim_{n \to \infty} \frac{a_{n}}{b_{n}} = \lim_{n \to \infty} \frac{\frac{n}{n^{2} + 1}}{\frac{1}{n}} = \lim_{n \to \infty} \frac{n^{2}}{n^{2} + 1} = \lim_{n \to \infty} \frac{1}{1 + \frac{1}{n^{2}}} = \frac{1}{1 + 0} = 1 \Rightarrow \lim_{n \to \infty} \frac{a_{n}}{b_{n}} = 1 and ∵ \sum_{n = 1}^{\infty} \frac{1}{n} is divergent by p-series test. ∴ by limit comparison test \sum_{n = 1}^{\infty} \frac{1}{n + \frac{1}{n}} is divergent.$

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: Structure And Method, Book 1

Algebra

ISBN:

9780395977224

Author:

Richard G. Brown, Mary P. Dolciani, Robert H. Sorgenfrey, William L. Cole

Publisher:

McDougal Littell