Find the radius of convergence and interval of convergence...

Key Concepts

Radius of Convergence

The radius of convergence is the non-negative number R such that the power series converges absolutely for |x - c| < R and diverges for |x - c| > R. It is determined using methods like the ratio test or the root test, which assess the limit behavior of the series' coefficients.

Ratio Test

The ratio test is a common method used to determine absolute convergence of a series. For a series ?a?, it involves taking the limit of |a???/a?| as n approaches infinity. In the context of power series, this limit can be expressed in terms of |x - c| and leads directly to finding the radius of convergence by setting the limit less than 1.

Power Series

A power series is an infinite series of the form ? a? (x ? c)?, where a? represents the coefficients and c is the center of the series. It represents functions as infinite sums and its convergence depends on the value of x relative to the series’ center.

Interval of Convergence

The interval of convergence is the set of x-values for which the power series converges. It is centered at c and has a length of 2R, with the endpoints possibly requiring separate testing to determine whether the series converges or diverges at those points.

Transcript

00:01 We want to find the radius of convergence and interval of convergence of the power series from and equals 1 to infinity of x to the nth over n to the 4th times 4 to the nth.

00:19 And so we can apply the ratio test here.

00:22 If we let l equal the limit as n approaches infinity of the absolute value of the next term over the current term, this would be equal to the limit as n approaches infinity of.

00:44 The next term would be defined as x to the n plus 1 over n plus 1 to the 4th times 4 to the n plus 1.

01:02 And this would be all over the current term x to the nth over end of the 4th times 4 to the nth.

01:12 All that within the opposite value, which this would be equal to the limit.

01:18 As n approaches infinity of the absolute value of x to the n plus 1 over n plus 1 times 4 to the n plus 1 times again this is the ratio test n to the 4th times 4 to the n over the absolute value of x to the nth and this would be equal to the limit as n approaches infinity of the absolute value of x over 4 times n over n plus 1 to the 4th.

01:58 We divide the numerator and denominator by n.

02:04 We have the limit as n approaches infinity of the absolute value of x over 4 times n over n over 1 over n plus n over n.

02:21 Again we're dividing everything by n in the numerator and denominator.

02:27 And this is to the fourth.

02:33 So now we have the limit as n approaches infinity of the absolute value of x over 4 times 1 over 1 over n plus 1 to the 4 which would be equal to the absolute value of x over 4 times if we approach infinity that would be 1 over 0 plus 1 to the 4th...

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