Problem 18 - Essential Calculus Early Transcendentals

Question

Find the radius of convergence and interval of convergence of the series. $$\sum_{n=1}^{\infty} \frac{n}{4^{n}}(x+1)^{n}$$

Step-by-Step Solution

Answer

The radius of convergence is 4, and the interval of convergence is \((-5, 3)\).

Step 1: Identify the Series

The given series is \( \sum_{n=1}^{\infty} \frac{n}{4^{n}}(x+1)^{n} \). This is a power series centered at \( x = -1 \).

Step 2: Use the Ratio Test for Radius of Convergence

The Ratio Test involves taking the limit of the absolute value of the ratio of consecutive terms. For our series, the \( n^{th} \) term is \( a_n = \frac{n}{4^{n}}(x+1)^{n} \) and the \( (n+1)^{th} \) term is \( a_{n+1} = \frac{n+1}{4^{n+1}}(x+1)^{n+1} \).

Step 3: Apply the Ratio

Compute \( \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = \lim_{n \to \infty} \left| \frac{(n+1)}{4^{n+1}} (x+1) \frac{4^n}{n} \right| = \frac{|x+1|}{4} \lim_{n \to \infty} \frac{n+1}{n} \).

Step 4: Evaluate the Limit

Simplify the expression to \( \lim_{n \to \infty} \left| \frac{(n+1)}{n} \right| = 1 \). Thus, the limit becomes \( \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = \frac{|x+1|}{4} \).

Step 5: Find the Radius of Convergence

For convergence, we require \( \frac{|x+1|}{4} < 1 \). Solving this inequality gives \( |x+1| < 4 \). So, the radius of convergence \( R \) is 4.

Step 6: Determine the Interval of Convergence

Since \( |x+1| < 4 \), we can write \( -4 < x+1 < 4 \), which simplifies to \( -5 < x < 3 \). The interval of convergence will need to consider the endpoints separately.

Step 7: Test Endpoint x = -5

Substitute \( x = -5 \) into the original series to get \( \sum_{n=1}^\infty \frac{n}{4^n}(-4)^{n} \), which simplifies to \( \sum_{n=1}^\infty (-1)^n n \). This series diverges by the nth term test.

Step 8: Test Endpoint x = 3

Substitute \( x = 3 \) into the original series to get \( \sum_{n=1}^\infty \frac{n}{4^n} 4^n \), which simplifies to \( \sum_{n=1}^\infty n \), and this series also diverges by the nth term test.

Step 9: Conclude the Interval of Convergence

Since both endpoints \( x = -5 \) and \( x = 3 \) do not converge, the interval of convergence is \( (-5, 3) \).

Key Concepts

Interval of Convergence

The interval of convergence of a power series indicates the set of values for which the series converges. For the series in question, we found that the radius of convergence is 4. This means the series will converge within a distance of 4 from the center, which is at -1.So, the series converges when a\(|x+1| < 4\). Translating this inequality to an interval gives:ul>

Starting at -1 - 4 = -5

Ending at -1 + 4 = 3

The interval of convergence is To ensure the interval's accuracy, we need to check if the series converges at the endpoints -5 and 3. In this case, both endpoints diverge, and hence the interval is (-5, 3).Identifying the interval is crucial, as functions within this interval behave consistently, allowing calculus operations like differentiation and integration.

Power Series

A power series is a way of expressing a function as an infinite sum of terms. Each term in the series is usually a product of a constant coefficient and a power of the variable. The general form of a power series centered at a point c is:\[\sum_{n=0}^{\infty} a_n (x-c)^n\]In our exercise, the given power series is:\[\sum_{n=1}^{\infty} \frac{n}{4^{n}}(x+1)^{n}\]Let's break it down:

• Coefficients a_n = \frac{n}{4^n
• Center c= -1
• Variable (x+1)^n

In such series, the focus is on understanding how the series behaves for different values of x. Within the interval of convergence, the power series represents a valid function, often making it useful for approximations or solving differential equations.

Ratio Test

The ratio test is a method used to find the convergence of a power series. It involves looking at the ratio of consecutive terms of the series. Applying the ratio test helps in finding the radius of convergence, which describes for what values the series converges.For the series \(a_n = \frac{n}{4^n}(x+1)^n\), the ratio of \(a_{n+1}\) to \(a_n\) is:\[\lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = \lim_{n \to \infty} \left| \frac{(n+1)}{4^{n+1}}(x+1)\frac{4^n}{n}\right| = \frac{|x+1|}{4}\]Simplifying gives \(\lim_{n \to \infty} \frac{n+1}{n} = 1\), leaving us with\(\frac{|x+1|}{4}\).For convergence:

• The limit must be less than 1, which divides our work in determining the radius of convergence.
• For this problem, that results in\(|x+1| < 4\), so the radius of convergence is 4.

This test is extremely helpful for quickly finding if a series converges and under what conditions.

Nth Term Test

The nth term test provides us with a straightforward method to determine the divergence of a series. Essentially, if the limit of the nth term of a series as \(n\) approaches infinity does not equal zero, the series diverges.In our exercise, when finding the interval of convergence, it is essential to check series convergence at the boundaries by plugging the endpoints back into the original series. Let's see:

• For \(x = -5\), the transformed series is \(\sum_{n=1}^{\infty} (-1)^n n\), where terms do not tend to zero. Hence, it diverges.
• For \(x = 3\)

It's crucial to remember that while a limit of zero does not guarantee convergence (further tests can be needed), a non-zero limit certainly means divergence. Therefore, the nth term test helped conclude that the external bounds are not part of the interval of convergence in this particular case.