Question

A power series is given. (a) Find the radius of convergence. (b) Find the interval of convergence. $$\sum_{n=0}^{\infty} \sqrt{n} x^{n}$$

Short answer

Radius: 1, Interval: \((-1,1)\).

Step-by-step solution

Apply the Root Test

To find the radius of convergence, we apply the Root Test to the series \( \sum_{n=0}^{\infty} \sqrt{n} x^{n} \). The Root Test involves finding \( \lim_{n \to \infty} \sqrt[n]{|a_n|} \), where \( a_n = \sqrt{n} x^n \). This simplifies to \( \lim_{n \to \infty} \sqrt[n]{\sqrt{n} x^n} = \lim_{n \to \infty} \left(n^{1/(2n)}\right) |x| \). Since \( n^{1/(2n)} \to 1 \) as \( n \to \infty \), the expression further simplifies to \( |x| \).

Determine Radius of Convergence

Set the result of the Root Test less than 1 for convergence: \( |x| < 1 \). This tells us that the radius of convergence \( R \) is 1, which means \( x \) must be within 1 unit from 0 for the series to converge.

Establish Interval of Convergence

The radius tells us that the series converges for \( -1 < x < 1 \). Next, we check the endpoints. For \( x = 1 \), the series becomes \( \sum_{n=0}^{\infty} \sqrt{n} \), which diverges.For \( x = -1 \), the series becomes \( \sum_{n=0}^{\infty} \sqrt{n} (-1)^n \), which also diverges (using the Test for Divergence as the terms do not approach zero). Therefore, the series converges only in the open interval \((-1, 1)\).

Conclusion

With the calculations and tests of the series at the endpoints, we conclude that the interval of convergence is \((-1, 1)\), and the radius of convergence is 1.

Key concepts

Radius of Convergence

The radius of convergence is a crucial concept in understanding the behavior of power series. It determines how far from the center of the series, usually denoted as zero, the series will converge. In the exercise involving the series \( \sum_{n=0}^{\infty} \sqrt{n} x^{n} \), we utilize the Root Test to find this radius.
The general strategy for finding the radius of convergence involves evaluating the limit \( \lim_{n \to \infty} \sqrt[n]{|a_n|} \), where \( a_n \) represents the series terms. For our specific series, plugging in \( a_n = \sqrt{n} x^n \) and simplifying gives us \( |x| \) in the limit. When \( |x| < 1 \), the series converges absolutely. This leads directly to the radius of convergence \( R = 1 \), indicating that the series converges when \( x \) is within 1 unit of 0.

Interval of Convergence

Once the radius of convergence is known, determining the interval of convergence helps us understand the exact range of \( x \) values for which the series converges. The interval of convergence includes all \( x \) values that make the power series converge and requires examining the behavior of the series at the endpoints of the interval suggested by the radius.
For the series \( \sum_{n=0}^{\infty} \sqrt{n} x^{n} \), the radius tells us that the series converges for \( -1 < x < 1 \). We then must check the endpoints, \( x = 1 \) and \( x = -1 \). By substituting these values into the series, we evaluate the series' behavior. At both endpoints, the series diverges. Therefore, the interval of convergence is \((-1, 1)\), excluding the endpoints themselves.

Root Test

The Root Test is a handy tool for determining convergence in series, especially power series. It involves taking the \( n \)-th root of the absolute value of a series term and analyzing its limit as \( n \) approaches infinity. If the limit \( L = \lim_{n \to \infty} \sqrt[n]{|a_n|} \) is less than 1, the series converges absolutely; if greater than 1, it diverges; if exactly 1, the test is inconclusive.
In the series \( \sum_{n=0}^{\infty} \sqrt{n} x^{n} \), using the Root Test involves simplifying \( \lim_{n \to \infty} \sqrt[n]{\sqrt{n} x^n} \) to \( |x| \) as \( n^{1/(2n)} \) approaches 1. Since \( |x| < 1 \) for convergence, this reinforces the understanding of why the radius of convergence is 1. The Root Test offers a clear pathway to identifying the values of \( x \) for which the series behaves nicely, aiding in determining both the radius and interval of convergence.