In Problems 1-4, find the radius of convergence and interval of convergence for the given power

In Problems 1-4, find the radius of convergence and interval...

Key Concepts

Endpoint Analysis

Endpoint analysis is the process used to determine the behavior of a power series at the boundary points of its interval of convergence. Even after finding a radius of convergence, each endpoint must be tested individually with appropriate convergence tests, as the Ratio Test does not provide information at these specific points.

Ratio Test

The Ratio Test is a common method used to determine the convergence of infinite series, particularly useful for power series. It involves taking the limit of the absolute value of the ratio of successive terms; if this limit is less than one, the series converges absolutely.

Interval of Convergence

The interval of convergence is the set of all x-values for which the power series converges. After calculating the radius of convergence, the endpoints of the interval must be examined separately, as the series may converge, diverge, or converge conditionally at those points.

Radius of Convergence

The radius of convergence is the value that defines the distance from the center of the power series within which the series converges absolutely. It is determined using methods like the Ratio Test or the Root Test and indicates the range of x-values for which the series behaves well.

Power Series

A power series is an infinite sum of terms involving powers of a variable (usually written as (x - c)^n) multiplied by coefficient sequences. It serves as a tool for representing functions and analyzing their behavior over intervals of convergence.

Transcript

00:01 Hi guys, in this problem we need to find the radius of conversions and interval of conversions for this given power series, where the power series is summation over n from 1 to infinity for 2 power n over n times x power n.

00:23 Okay, so let this power series is equal to u n.

00:29 Okay.

00:30 So now we need to find limit where n goes to infinity for modulus of u of n plus 1 over u n.

00:44 Okay, so this is equal to limitation or limit of n where goes to infinity over u sub n plus 1 is 2 power n plus 1.

01:06 Times x power n plus 1 over n plus 1 over here we have 2 power n times x power n over n okay so this can be 3 written as limit when n goes to infinity okay for a modulus of 2 power n plus 1 times x power n plus 1 over n plus 1 times n over 2 power n times x power n okay so this is equal to limit when n goes to infinity for a modulus of 2 x n over n plus 1 or 2 x over 1 plus 1 or 2 x over 1 plus 1 over n over n okay, so this is equal to modulus 2x.

02:18 By the ratio test, the series converts for modulus 2x less than one and diverged for modulus 2x more than one.

02:29 So the series converges for modulus x less than 1 over 2.

02:43 Okay, so in diversion otherwise.

02:47 So the radius of conversions for this series here let the radius be r so the radius here is 1 over 2 because the series is centered at x equals 0 it will converge in the open interval from negative 1 over 2 to 1 over 2 and sorry this is open interval not close interval okay okay, now we need to determine whether the series converged at the endpoints or not.

03:28 Okay, so we need to substitute 1 over 2 for x in the given power series...

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