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The radius of convergence of power series Sigma n=0 to infinity z^n/n is 1. True or False?
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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.
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The radius of convergence of power series Sigma n=0 to infinity z^n/n is 1. True or False?
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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?.
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