Q. 4 Use Exercise 3 to explain why th... [FREE SOLUTION]

Chapter 7: Q. 4 (page 639)

Use Exercise 3 to explain why the ratio test will be inconclusive for every series ${鈭�}_{k = 1}^{鈭�} a_{k}$ in which $a_{k}$ is a polynomial.

Short Answer

Expert verified

If $a_{k}$ is a polynomial then localid="1649104353595" $\lim_{x 鈫� 鈭�} \frac{p (x + 1)}{p (x)}$ will be $1$ and the ratio test is inconclusive for series when $\lim_{x 鈫� 鈭�} \frac{p (x + 1)}{p (x)} = 1 .$

Step by step solution

Step 1. Given information.

If $p (x)$ is a polynomial then role="math" localid="1649104261634" $\lim_{x 鈫� 鈭�} \frac{p (x + 1)}{p (x)} = 1 .$

If $a_{k}$ is a polynomial then the ratio test will be inconclusive for every series role="math" localid="1649104105405" ${鈭�}_{k = 1}^{鈭�} a_{k} .$

Step 2. Verification.

Consider $a_{k} = p (x)$

role="math" localid="1649104033456" $a_{k + 1} = p (k + 1) \lim_{k 鈫� 鈭�} \frac{a_{k + 1}}{a_{k}} = \lim_{k 鈫� 鈭�} \frac{p (k + 1)}{p (k)}$

as $\lim_{k 鈫� 鈭�} \frac{p (k + 1)}{p (k)} = 1$

so role="math" localid="1649104148331" $\lim_{k 鈫� 鈭�} \frac{a_{k + 1}}{a_{k}} = 1 L = 1$

According to the ratio test, if ${鈭�}_{k = 1}^{鈭�} a_{k}$ is a series with positive terms and $L = \lim_{k 鈫� 鈭�} \frac{a_{k + 1}}{a_{k}} = 1$ then then the ratio test will be inconclusive for series.

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

Use Exercise 3 to explain why the ratio test will be inconclusive for every series ${鈭�}_{k = 1}^{鈭�} a_{k}$ in which $a_{k}$ is a polynomial.

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Verification.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Geometry

Calculus

Discrete Mathematics

Statistics

Decision Maths

Study anywhere. Anytime. Across all devices.

91影视

Use Exercise 3 to explain why the ratio test will be inconclusive for every series 鈭�k=1鈭�akin which akis a polynomial.

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Verification.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Geometry

Calculus

Discrete Mathematics

Statistics

Decision Maths

Study anywhere. Anytime. Across all devices.

Use Exercise 3 to explain why the ratio test will be inconclusive for every series ${鈭�}_{k = 1}^{鈭�} a_{k}$ in which $a_{k}$ is a polynomial.