Question

A power series is given. (a) Find the radius of convergence. (b) Find the interval of convergence. $$\sum_{n=0}^{\infty} 5^{n}(x-1)^{n}$$

Short answer

(a) The radius of convergence is \( \frac{1}{5} \). (b) The interval of convergence is \((0.8, 1.2)\).

Step-by-step solution

Identify the power series

The given power series is \( \sum_{n=0}^{\infty} a_n (x-c)^{n} \), where \( a_n = 5^n \) and \( c = 1 \). The general term is \( a_n (x-1)^n \).

Use the Ratio Test for convergence

To find the radius of convergence, we use the Ratio Test. Let \( a_n = 5^n \). We calculate:\[ \frac{a_{n+1}}{a_n} = \frac{5^{n+1}}{5^n} = 5. \] Then apply the Ratio Test:\[ \lim_{n \to \infty} \left| \frac{a_{n+1} (x-1)^{n+1}}{a_n (x-1)^n} \right| = \lim_{n \to \infty} \left| (x-1) \right| \cdot 5. \]

Solve for convergence condition

For convergence, we need \[ \left| (x-1) \right| \cdot 5 < 1. \]Solving this inequality gives\[ \left| x-1 \right| < \frac{1}{5}. \] Thus, the radius of convergence \( R \) is \( \frac{1}{5} \).

Determine the interval of convergence

The center of the series is \(1\), and the radius is \( \frac{1}{5} \). Therefore, the interval of convergence is initially \((1 - \frac{1}{5}, 1 + \frac{1}{5}) = (0.8, 1.2)\).

Test the endpoints of the interval

Test \( x = 0.8 \): The series becomes \( \sum_{n=0}^{\infty} 5^n (-0.2)^n \), which is a geometric series with ratio greater than 1 and hence diverges.Test \( x = 1.2 \): The series becomes \( \sum_{n=0}^{\infty} 5^n (0.2)^n \), which likewise diverges. Thus, the endpoints do not contribute to the convergence.

Key concepts

Radius of Convergence

When dealing with power series, a fundamental concept is the radius of convergence. This tells us how far from the center, or point of expansion, our series converges. In the given power series \( \sum_{n=0}^{\infty} 5^{n}(x-1)^{n} \), the goal is to determine this radius.

We use the Ratio Test as a standard method to find the radius of convergence \( R \). For our series, the general term is simplified by focusing on the part of the series with \( 5^n \). The Ratio Test involves taking the limit:

• Calculate the ratio of consecutive terms: \( \frac{a_{n+1}}{a_n} = 5 \).
• Apply the Ratio Test: \( \lim_{n \to \infty} \left| \frac{a_{n+1} (x-1)^{n+1}}{a_n (x-1)^n} \right| = \left| (x-1) \right| \cdot 5 \).

To satisfy the convergence condition, the limit must be less than 1. This results in \( \left| x-1 \right| < \frac{1}{5} \). Hence, the radius of convergence is \( \frac{1}{5} \). This means the power series will converge for all \( x \) within this radius around \( x = 1 \).

Interval of Convergence

Once the radius of convergence is known, the next step is to find the interval of convergence. This interval indicates the specific range of \( x \) values for which the power series converges.

For our series, since the radius of convergence is \( \frac{1}{5} \), the interval is centered at \( x = 1 \) with endpoints determined by the radius:

• Start by finding the open interval: \( (1 - \frac{1}{5}, 1 + \frac{1}{5}) = (0.8, 1.2) \).
• Check behavior at endpoints by substituting \( x = 0.8 \) and \( x = 1.2 \).

Testing the endpoints ensures accuracy:- At \( x = 0.8 \), the series turns into a geometric series whose ratio is greater than 1, thus it diverges.- Similarly, at \( x = 1.2 \), the series continues to diverge.

Since the series diverges at both endpoints, they are not included in the interval. Hence, the interval of convergence is \( (0.8, 1.2) \). This is the precise range where the series will converge.

Ratio Test

The Ratio Test is a powerful tool used to determine the convergence of series. It is specifically effective when dealing with power series like \( \sum_{n=0}^{\infty} 5^{n}(x-1)^{n} \).

The procedure involves analyzing the limit of the ratio of the absolute values of consecutive terms of the series. Here's how it applies to our series:

• Calculate \( \frac{a_{n+1} (x-1)^{n+1}}{a_n (x-1)^n} \) to get \( 5 \cdot |x-1| \).
• Determine convergence by ensuring \( \lim_{n \to \infty} |5(x-1)| < 1 \), which simplifies to \( |x-1| < \frac{1}{5} \).

Thus, the Ratio Test gives us the condition necessary for convergence: \( x \) must lie within \( \frac{1}{5} \) of 1.

Utilizing the Ratio Test simplifies our process to find both the radius and interval of convergence. It's particularly useful because it accommodates the geometric growth factor \( 5^n \) seen in our series, pinpointing exactly when and where the series will behave properly.