UPSC Exam > UPSC Questions > The radius of convergence of power series Sig...
Start Learning for Free
The radius of convergence of power series Sigma n=0 to infinity z^n/n is 1. True or False?
Most Upvoted Answer
The radius of convergence of power series Sigma n=0 to infinity z^n/n ...
Radius of Convergence of Power Series
False
Explanation:
Definition of Radius of Convergence:
The radius of convergence of a power series is the distance from the center of the series to the nearest point where the series converges.
Given Power Series:
The power series in question is Σ z^n/n, where n ranges from 0 to infinity.
Calculating the Radius of Convergence:
To find the radius of convergence, we can use the ratio test. Let's apply the ratio test to the given series:
r = lim(n→∞) |a(n+1)/a(n)|
= lim(n→∞) |z^(n+1)/(n+1)| / |z^n/n|
= lim(n→∞) |z/(n+1)|
= 0
Since the limit of the ratio is 0, the radius of convergence is infinite. This means that the power series Σ z^n/n converges for all values of z.
Therefore, the statement that the radius of convergence of the power series Σ z^n/n is 1 is False.
False
Explanation:
Definition of Radius of Convergence:
The radius of convergence of a power series is the distance from the center of the series to the nearest point where the series converges.
Given Power Series:
The power series in question is Σ z^n/n, where n ranges from 0 to infinity.
Calculating the Radius of Convergence:
To find the radius of convergence, we can use the ratio test. Let's apply the ratio test to the given series:
r = lim(n→∞) |a(n+1)/a(n)|
= lim(n→∞) |z^(n+1)/(n+1)| / |z^n/n|
= lim(n→∞) |z/(n+1)|
= 0
Since the limit of the ratio is 0, the radius of convergence is infinite. This means that the power series Σ z^n/n converges for all values of z.
Therefore, the statement that the radius of convergence of the power series Σ z^n/n is 1 is False.
Attention UPSC Students!
To make sure you are not studying endlessly, EduRev has designed UPSC study material, with Structured Courses, Videos, & Test Series. Plus get personalized analysis, doubt solving and improvement plans to achieve a great score in UPSC.
|
Explore Courses for UPSC exam
|
|
Top Courses for UPSCView all
The radius of convergence of power series Sigma n=0 to infinity z^n/n is 1. True or False?
Question Description
The radius of convergence of power series Sigma n=0 to infinity z^n/n is 1. True or False? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about The radius of convergence of power series Sigma n=0 to infinity z^n/n is 1. True or False? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The radius of convergence of power series Sigma n=0 to infinity z^n/n is 1. True or False?.
The radius of convergence of power series Sigma n=0 to infinity z^n/n is 1. True or False? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about The radius of convergence of power series Sigma n=0 to infinity z^n/n is 1. True or False? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The radius of convergence of power series Sigma n=0 to infinity z^n/n is 1. True or False?.
Solutions for The radius of convergence of power series Sigma n=0 to infinity z^n/n is 1. True or False? in English & in Hindi are available as part of our courses for UPSC.
Download more important topics, notes, lectures and mock test series for UPSC Exam by signing up for free.
Here you can find the meaning of The radius of convergence of power series Sigma n=0 to infinity z^n/n is 1. True or False? defined & explained in the simplest way possible. Besides giving the explanation of
The radius of convergence of power series Sigma n=0 to infinity z^n/n is 1. True or False?, a detailed solution for The radius of convergence of power series Sigma n=0 to infinity z^n/n is 1. True or False? has been provided alongside types of The radius of convergence of power series Sigma n=0 to infinity z^n/n is 1. True or False? theory, EduRev gives you an
ample number of questions to practice The radius of convergence of power series Sigma n=0 to infinity z^n/n is 1. True or False? tests, examples and also practice UPSC tests.
|
|
Explore Courses for UPSC exam
|
|
Signup for Free!
Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests.
|
© EduRev
|
Education Revolution
|
Follow Us
|
Signup to see your scores
go up within 7 days!
Access 1000+ FREE Docs, Videos and Tests
Takes less than 10 seconds to signup