Question

A power series is given. (a) Find the radius of convergence. (b) Find the interval of convergence. $$\sum_{n=0}^{\infty}(-1)^{n} n !(x-10)^{n}$$

Short answer

The radius of convergence is 0, and the interval of convergence is \( x = 10 \).

Step-by-step solution

Understand the Formula for the Radius of Convergence

To find the radius of convergence, we'll use the formula for the radius of convergence of a power series \( \sum a_n (x - c)^n \): it is found using the ratio test \( R = \frac{1}{\limsup_{n \to \infty} \sqrt[n]{|a_n|}} \) or the root test. For this power series, \( a_n = (-1)^n n! \).

Apply the Root Test

The root test is often effective with factorials. Compute \( \limsup_{n \to \infty} \sqrt[n]{|n!|} \). Since \( \sqrt[n]{n!} \approx \frac{n}{e} \) when \( n \) is large, the limsup is \( \infty \). Thus, \( \limsup_{n \to \infty} \sqrt[n]{|n!|} = \infty \).

Calculate the Radius of Convergence

Using the result from the root test, the radius of convergence \( R = \frac{1}{\infty} = 0 \). Thus, the power series converges only when \( x = 10 \).

Determine the Interval of Convergence

Since the radius of convergence is 0, the interval of convergence is simply the point \( x = 10 \). In this case, the series converges only at this exact value of \( x \).

Key concepts

Interval of Convergence

In the realm of power series, the interval of convergence is a crucial concept. It is the set of all values of \( x \) for which the series converges. Once we determine the radius of convergence \( R \), we can typically find the interval by examining the series in the form \( |x - c| < R \). Here, \( c \) is the center of the series, which is given as 10 in the exercise. However, in scenarios where \( R = 0 \), such as in this exercise, the series converges only at the center, \( x = 10 \). This simply means there is no interval, just a single point. This happens when the coefficients grow too rapidly or involve factorial expressions, causing convergence at one specific point only. Understanding these limitations helps us better interpret the behavior of power series. Remember:

• If \( R > 0 \), the series converges on an interval.
• If \( R = \infty \), there is convergence for all real numbers.
• If \( R = 0 \), it converges at the center point alone.

Root Test

The root test is an essential tool in determining convergence of a power series, especially when dealing with factorial terms. The root test assesses the limit\[\limsup_{n \to \infty} \sqrt[n]{|a_n|}\]which assists in calculating the radius of convergence \( R \). For a given series \( \sum a_n (x - c)^n \), apply the root test by examining the behavior of the sequence \( |a_n| \). In our exercise, the terms \( a_n = (-1)^n n! \) include factorials, which increase very quickly. Using the approximation \( \sqrt[n]{n!} \approx \frac{n}{e} \) for large \( n \), we determine that this limit approaches infinity. This result implies that the radius of convergence is \( R = \frac{1}{\infty} = 0 \).The root test effectively tells us where the series can converge based on the nature of its coefficients. It is especially powerful when factorials or exponential growth is involved. Always remember:

• The test requires the term root \( \, \sqrt[n]{|a_n|} \), making it suitable for large terms.
• It gives conclusive results when the limit is infinity or a specific value.

Power Series

Power series are expressions in the form \( \sum_{n=0}^{\infty} a_n (x - c)^n \), where \( c \) is the center. They are infinite in nature and extend the concept of polynomials by allowing an unlimited number of terms. Power series can represent functions within their interval of convergence and are widely used in calculus for function approximation.The power series in this exercise: \( \sum_{n=0}^{\infty} (-1)^n n!(x-10)^n \) is centered around \( x = 10 \). The presence of factorial terms \( n! \) induces rapid growth of the coefficients \( a_n \). This characteristic largely influences convergence behavior, limiting the series convergence solely to the center at \( x = 10 \).Key attributes of power series:

• They can approximate functions for values within the interval of convergence.
• The radius of convergence \( R \) also indicates where the series is usable.
• The behavior at end points often requires separate testing.

Power series are foundational in mathematical analysis and play a role in diverse fields like physics and engineering.