Question

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

Short answer

(a) Radius of convergence is 1; (b) Interval of convergence is \((-1, 1)\).

Step-by-step solution

Identify the Power Series

The given power series is \( \sum_{n=0}^{\infty} n x^n \). We need to find its radius of convergence and interval of convergence.

Use the Ratio Test

To find the radius of convergence, we use the Ratio Test. Consider the general term \( a_n = n x^n \). The Ratio Test requires calculating \( \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| \).

Compute Ratio of Successive Terms

Start by finding \( a_{n+1} = (n+1) x^{n+1} \). Thus, the ratio is \( \frac{a_{n+1}}{a_n} = \frac{(n+1)x^{n+1}}{n x^n} = \frac{n+1}{n} \cdot x \).

Find the Limit of the Ratio

Now compute the limit: \( \lim_{n \to \infty} \left| \frac{n+1}{n} \cdot x \right| = \lim_{n \to \infty} \left| \left(1 + \frac{1}{n}\right)x \right| = |x| \).

Determine the Radius of Convergence

For convergence, the limit must be less than 1: \( |x| < 1 \). Thus, the radius of convergence \( R = 1 \).

Check the Endpoints of the Interval

Since the radius of convergence is 1, we check the interval \([-1, 1]\). If \(x = 1\), the series becomes \(\sum_{n=0}^{\infty} n\), divergent. If \(x = -1\), the series becomes \(\sum_{n=0}^{\infty} (-1)^n n\), also divergent.

Conclude the Interval of Convergence

Since both endpoints lead to divergence, the interval of convergence is \((-1, 1)\).

Key concepts

Radius of Convergence

When dealing with power series, the radius of convergence is a crucial concept. It tells us the range for the variable, often denoted as \( x \), within which the series will converge. For this exercise, we are dealing with the power series \( \sum_{n=0}^{\infty} n x^n \). To find the radius of convergence, we utilize the Ratio Test. This involves calculating the limit of the absolute value of the ratio of successive terms in the series as \( n \to \infty \).
For the given series, the general term is \( a_n = n x^n \). We then consider the term \( a_{n+1} = (n+1) x^{n+1} \). By forming the ratio \( \frac{a_{n+1}}{a_n} = \frac{(n+1)x^{n+1}}{n x^n} \), and simplifying it to \( \frac{n+1}{n} \cdot x \), we proceed to find its limit.
The limit of the ratio \( \lim_{n \to \infty} \left| \frac{n+1}{n} \cdot x \right| \) comes out to be \(|x|\). For the series to converge, this limit should be less than 1. Thus, the radius of convergence, denoted by \( R \), is found to be 1. The radius of convergence indicates that the series will converge when \(-1 < x < 1\).

Interval of Convergence

The interval of convergence relates closely to the radius of convergence. It defines the actual values of \( x \) where the series converges completely. Knowing the radius is just part of the story; we must also check the endpoints to determine the full interval.
In our example of \( \sum_{n=0}^{\infty} n x^n \), with a radius of convergence \( R = 1 \), we initially determine the theoretical interval as \([-1, 1]\). However, it is crucial to evaluate convergence at the boundaries \( x = -1 \) and \( x = 1 \).
Upon testing the endpoints, at \( x = 1 \), the series \( \sum_{n=0}^{\infty} n \) diverges because it represents an arithmetic series with no limiting sum. Similarly, at \( x = -1 \), the series \( \sum_{n=0}^{\infty} (-1)^n n \) alternates but does not converge. Therefore, neither endpoint belongs in the interval of convergence.
Consequently, the interval of convergence for this power series is \((-1, 1)\), excluding the endpoints. This tells us where the series definitively adds up to some finite value and where our analysis yields reliable results.

Ratio Test

The Ratio Test is a powerful tool for determining the convergence or divergence of series. It is particularly useful for series where terms contain expressions raised to the power of n, such as our power series here. By examining the behavior of the ratio of successive terms, we can gain insights into the convergence properties of the series.
The test requires us to find \( \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| \). For our power series \( \sum_{n=0}^{\infty} n x^n \), we calculated this to be \( \lim_{n \to \infty} \left| \frac{n+1}{n} \cdot x \right| = |x| \). According to the Ratio Test, a series converges if this limit is less than 1. This provides not only the radius of convergence but also a starting point for determining the interval of convergence.
Thus, through the Ratio Test, we establish the fundamental convergence condition \( |x| < 1 \). This test helps delineate when the series transitions from converging to diverging as \( x \) varies, and is one of the cornerstones of analyzing power series.