Find the radius of convergence and interval of convergence of the series. ∑n = 1^∞ (√(n))/(8^n)

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

Key Concepts

Ratio Test

The ratio test is a common method for determining the convergence of series, particularly power series. By examining the limit of the absolute value of the ratio of successive terms, one can establish conditions on x that guarantee convergence, which directly leads to the determination of the radius of convergence.

Endpoint Analysis

Endpoint analysis involves checking the convergence of a power series at the boundary points of its interval of convergence. Since the standard convergence tests often provide information only on absolute convergence within the open interval, each endpoint must be individually tested using appropriate convergence tests to ascertain whether the series converges or diverges at those points.

Radius of Convergence

The radius of convergence is the distance from the center of the power series within which the series converges absolutely. Determining this radius is crucial for understanding where the series represents a valid function, and it is often found using tests such as the ratio or root test.

Power Series

A power series is an infinite series in the form ? a_n (x - c)^n, where each term involves a power of (x - c) multiplied by a coefficient. This representation is fundamental because it allows one to express functions as sums of infinite terms and analyze the behavior of the series depending on the value of x relative to the center c.

Interval of Convergence

The interval of convergence is the set of all x values for which the power series converges. It is defined by the radius of convergence and includes considerations of the endpoints, where convergence must be checked individually using other convergence tests.

Transcript

00:01 For the radius of convergence, we'll use the ratio test to figure out which values of x are allowed.

00:08 The ratio test will do limit as n goes to infinity of absolute value of a .n plus 1 over a .n.

00:18 Where a .n is this whole thing, including the x values.

00:25 Okay, so once you plug these things in, we get some cancellations to occur.

00:31 Would have a x plus 6 to the n plus 1 on top, but then we'd be dividing by x plus 6 to the n.

00:37 So we would just end up with the x plus 6.

00:41 And then we'd have an 8 to the n up top square root of n plus 1 over 8 to the n plus 1 square root of n.

00:52 Okay, so this is just from doing simple algebra here.

00:56 Remember our a .n terms of this whole thing, including the x values.

01:00 If we're dividing by a .n, the same thing as multiplying by the reciprocal.

01:04 So the reciprocal of this thing, we're going to have an aiden on top as we do.

01:08 And we're going to have a square root of n at the bottom as we do over here.

01:13 8 to the n divided by 8 to the n plus 1.

01:17 That's just going to turn into 1 eighth.

01:27 And then square root of n plus 1 divided by square root of n is square root of n plus 1 over n.

01:34 As n goes to infinity, n plus 1 over n is 1.

01:37 Square root of 1 is 1.

01:38 So this is just absolute value of x plus 6 over 8...

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