Question

A power series is given. (a) Find the radius of convergence. (b) Find the interval of convergence. $$\sum_{n=0}^{\infty} n^{2}\left(\frac{x+4}{4}\right)^{n}$$

Short answer

Radius of convergence: 4. Interval of convergence: \((-8, 0)\).

Step-by-step solution

Identify the General Term

The given power series is \( \sum_{n=0}^{\infty} n^{2}\left(\frac{x+4}{4}\right)^{n} \). The general term of this series can be written as \( a_n = n^2 \left(\frac{x+4}{4}\right)^n \).

Apply the Ratio Test

The ratio test helps in finding the radius of convergence. Consider the ratio \( \left| \frac{a_{n+1}}{a_n} \right| \). We compute:\[\left| \frac{a_{n+1}}{a_n} \right| = \frac{(n+1)^2\left(\frac{x+4}{4}\right)^{n+1}}{n^2\left(\frac{x+4}{4}\right)^n} = \left( \frac{n+1}{n} \right)^2 \cdot \left| \frac{x+4}{4} \right|\]Simplify the ratio:\[\frac{n^2 + 2n + 1}{n^2} \cdot \left| \frac{x+4}{4} \right| = \left( 1 + \frac{2}{n} + \frac{1}{n^2} \right) \cdot \left| \frac{x+4}{4} \right|\]

Take the Limit for the Ratio Test

The next step is to find the limit as \( n \to \infty \):\[\lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = \lim_{n \to \infty} \left( 1 + \frac{2}{n} + \frac{1}{n^2} \right) \cdot \left| \frac{x+4}{4} \right| = \left| \frac{x+4}{4} \right|\]For convergence, this limit must be less than 1: \( \left| \frac{x+4}{4} \right| < 1 \).

Solve for Radius of Convergence

The inequality \( \left| \frac{x+4}{4} \right| < 1 \) implies:\[ -1 < \frac{x+4}{4} < 1 \]Solve this compound inequality:\[-4 < x+4 < 4\]Subtract 4:\[-8 < x < 0\]Thus, the radius of convergence is 4, i.e., \( R = 4 \).

Determine the Interval of Convergence

To find the interval of convergence, test the endpoints \( x = -8 \) and \( x = 0 \).1. For \( x = -8 \), the series becomes \( \sum_{n=0}^{\infty} n^2 (-1)^n \). This series diverges because the terms do not cancel or diminish.2. For \( x = 0 \), the series is \( \sum_{n=0}^{\infty} n^2 \). This series also diverges as it grows without bound.Therefore, the interval of convergence is \( (-8, 0) \).

Key concepts

Understanding Power Series

A power series is essentially a sum of terms of the form \( a_n (x-c)^n \), where \( a_n \) represents the coefficients, \( x \) is the variable, and \( c \) is the center of the series. In our case, the power series given is \( \sum_{n=0}^{\infty} n^{2}\left(\frac{x+4}{4}\right)^{n} \). Here, each term in the series involves an expression that includes both a constant part and the variable raised to successive powers.
Power series are very useful because they can represent functions over certain intervals and help in evaluating complex expressions using relatively simple arithmetic operations.
When we deal with power series, we want to determine where this series converges to an actual number rather than diverging to infinity, which is called the interval of convergence.

Ratio Test for Convergence

To find where a power series converges, we often use the Ratio Test. The Ratio Test examines the limit of the ratio of successive terms. Suppose we have a general power series \( \sum_{n=0}^{\infty} a_n x^n \). The ratio test formula is used as follows:

• Set up the expression \( \left| \frac{a_{n+1}}{a_n} \right| \) to figure out how each term of the series relates to its predecessor.
• In this exercise, the ratio \( \left| \frac{a_{n+1}}{a_n} \right| = \left( \frac{n+1}{n} \right)^2 \cdot \left| \frac{x+4}{4} \right| \) is simplified to \( \left( 1 + \frac{2}{n} + \frac{1}{n^2} \right) \cdot \left| \frac{x+4}{4} \right| \).
• Find the limit of this ratio as \( n \to \infty \) to explore convergence behavior.

The power series converges if the limit of this ratio is less than 1.

Determining the Interval of Convergence

Once the radius of convergence is established, the next step is to determine the interval of convergence. This is the set of \( x \) values for which the series actually converges.
The expression \( \left| \frac{x+4}{4} \right| < 1 \) was simplified to find the values of \( x \):

• The inequality transforms to \( -1 < \frac{x+4}{4} < 1 \).
• This further simplifies to \( -8 < x < 0 \), identifying the core interval.

It's crucial to test the endpoints within this interval to check for their contribution to convergence. Here, since both endpoints \( x = -8 \) and \( x = 0 \) led to diverging series, neither was included in our final interval, leaving the interval of convergence as \( (-8, 0) \).

Solving the Inequality

To find the radius of convergence or interval of convergence, inequalities are often solved to find the range of \( x \). The reduction of complex inequality \( \left| \frac{x+4}{4} \right| < 1 \) is key:
The inequality can be rewritten and solved as two separate inequalities:

• \( -1 < \frac{x+4}{4} \) translates to \( \frac{x+4}{4} > -1 \), which further simplifies to \( x > -8 \).
• \( \frac{x+4}{4} < 1 \) leads to \( x < 0 \).

Combining these results, one finds \( -8 < x < 0 \), simplifying to a statement about the internal behavior of our original series.
These steps are not only essential for calculating the convergence radius but also for ensuring that every solution step is logically correct and clear.