Find the radius of convergence and interval of convergence of the series. ∑n = 1^∞ (( - 1)^n 4^n

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

Key Concepts

Convergence Tests

Convergence tests, including the Ratio Test, Root Test, and Alternating Series Test, are used to analyze the behavior of infinite series. These methods help determine whether a series converges or diverges by examining the limit of the sequence of terms or the absolute values of terms, thereby establishing criteria to find the radius and interval of convergence for power series.

Interval of Convergence

The interval of convergence of a power series is the set of all x-values for which the series converges, which includes the open interval (c - R, c + R) and possibly one or both endpoints. Determining the interval of convergence involves finding R as well as separately testing the endpoints to see if convergence occurs there using appropriate convergence tests such as the Alternating Series Test or the p-Series Test.

Radius of Convergence

The radius of convergence is a non-negative number R which defines the interval around the center of a power series within which the series converges absolutely. It is determined by applying tests such as the Ratio Test or the Root Test to the coefficients of the series. If |x - c| < R, the series converges absolutely, while if |x - c| > R, the series diverges.

Power Series

A power series is an infinite series of the form ? a? (x - c)?, where a? represents the coefficient of each term and c is the center of the series. Power series are used to represent functions as sums of infinitely many terms, and they play a key role in calculus, analysis, and differential equations due to their properties and ease of manipulation within their interval of convergence.

Transcript

00:01 Okay, so r as usual is limit as n goes to infinity, a -n over a -n plus 1, and it should be opposite values here.

00:12 The absolute value is going to get rid of this minus 1 to the n, because the a -n's here, by a -n we mean minus 1 to the n, times 4 -n divided by square root of n.

00:21 So these guys here, those are our a -ns.

00:25 So this is going to turn out to be limit as n goes to infinity of 4 to the n over square root of n.

00:35 And then dividing by a n plus 1 is the same thing as multiplying by the reciprocal.

00:40 So we get square root of n plus 1 divided by 4 to the n plus 1.

00:47 4 to the n divided by 4 to the n plus 1 is just 1 over 4.

00:55 We have 1 4th.

00:57 And then we can combine these square roots.

01:01 Limit as n goes to infinity of n plus 1 over n is 1.

01:05 So this guy turns into square root of 1, which is just 1.

01:10 And we're multiplying that by 1 4th.

01:12 So we get 1 4th as the radius of convergence.

01:16 For the interval of convergence, we just need to figure out whether or not we get convergence at 1 4th, and whether or not we get convergence at minus 1 4th.

01:25 Okay, so when x is minus 1 4th, if you plug in minus 1 4th into here, what you're going to end up getting is sum from n equals 1 .4th...

Please give Ace some feedback