USE THE RATIO TEST! Find the radius and interval of convergence ofseries below. Makecertain that you check the endpoints. State which test you are using and carefully outline howyou carried out the test to get your conclusion. Use the procedure outlined in class to get youranswers and credit for your work.(z-2)"(i) En=1 n+1’

Ratio test: The usual form of the test makes use of the limit {\displaystyle L=\lim _{n\to…

Answered: Find the radius and interv you check…

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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USE THE RATIO TEST!

Find the radius and interval of convergence of
series below. Make
certain that you check the endpoints. State which test you are using and carefully outline how
you carried out the test to get your conclusion. Use the procedure outlined in class to get your
answers and credit for your work.
(z-2)"
(i) En=1 n+1’

Expert Solution

Step 1

Ratio test:

The usual form of the test makes use of the limit

L=\lim _{n\to \infty }\left|{\frac {a_{n+1}}{a_{n}}}\right|.

The ratio test states that:

if L < 1 then the series converges absolutely;
if L > 1 then the series is divergent;
if L = 1 or the limit fails to exist, then the test is inconclusive, because there exist both convergent and divergent series that satisfy this case.

It is possible to make the ratio test applicable to certain cases where the limit L fails to exist, if limit superior and limit inferior are used. The test criteria can also be refined so that the test is sometimes conclusive even when L = 1. More specifically, let

R=\lim \sup \left|{\frac {a_{n+1}}{a_{n}}}\right|

r=\lim \inf \left|{\frac {a_{n+1}}{a_{n}}}\right|

Then the ratio test states that:

if R < 1, the series converges absolutely;
if r > 1, the series diverges;
if $\left|{\frac {a_{n+1}}{a_{n}}}\right|\geq 1$ for all large n (regardless of the value of r), the series also diverges; this is because $|a_{n}|$ is nonzero and increasing and hence $a n$ does not approach zero;
the test is otherwise inconclusive.

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