Problem 26 - Thomas Calculus

Question

In Exercises \(1-36,\) (a) find the series' radius and interval of convergence. For what values of \(x\) does the series converge (b) absolutely (c) conditionally? $$ \sum_{n=0}^{\infty} n !(x-4)^{n} $$

Step-by-Step Solution

Answer

The series converges only at \( x = 4 \) and is not conditionally convergent.

Step 1: Identify the Type of Series

The given series is \( \sum_{n=0}^{\infty} n!(x-4)^n \), which is a power series centered at \( x = 4 \). We will use the Ratio Test to find the radius of convergence.

Step 2: Apply the Ratio Test

For the Ratio Test, we look at the limit \( \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| \) where \( a_n = n!(x-4)^n \). We have \( a_{n+1} = (n+1)!(x-4)^{n+1} \). The ratio is:\[\left| \frac{(n+1)!(x-4)^{n+1}}{n!(x-4)^n} \right| = |x-4| \cdot (n+1).\]

Step 3: Find the Radius of Convergence

According to the Ratio Test, for convergence, \( \lim_{n \to \infty} |x-4| \cdot (n+1) < 1 \). However, since \( (n+1) \to \infty \), the limit will never be less than 1 for any finite \( x \), meaning the series diverges for all \( x eq 4 \). Therefore, the radius of convergence is zero.

Step 4: Determine the Interval of Convergence

Since the radius of convergence is zero, the interval of convergence is just \( x = 4 \). The series converges only at this single point.

Step 5: Analyze Absolute Convergence

Absolute convergence means the series \( \sum |a_n| \) converges. However, since the interval of convergence consists of only one point \( x = 4 \), it doesn't matter how \( x \) is substituted into \( |x-4| \), as only the point \( x=4 \) is possible. At this point, the series is 0 and it converges.

Step 6: Analyze Conditional Convergence

Conditional convergence would require that the series converges, but not absolutely. Since the series only converges at \( x = 4 \), and there it trivially converges to 0, the concept of conditional convergence does not apply in the classic sense.

Key Concepts

Power Series

A power series is a series of the form \( \sum_{n=0}^{\infty} a_n (x-c)^n \), where \( a_n \) are coefficients and \( c \) is the center of the series. The variable \( x \) can take different values, and depending on those values, the series might converge or diverge. A power series is a convenient way to represent functions, especially useful in calculus and mathematical analysis.

• Each term in the series is a power of \( (x-c) \).
• The series is centered at the point \( x = c \).
• The convergence behavior depends on the values of \( x \).

In the given series \( \sum_{n=0}^{\infty} n!(x-4)^n \), it is centered around \( x = 4 \). Understanding how the series behaves when \( x \) changes helps us solve the convergence problems.

Ratio Test

The ratio test is a method used to determine the convergence of a power series. It considers the limit of the ratio of absolute values of successive terms. The test uses the expression:\[\lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|,\]where \( a_n \) are the terms of the series.

• If the limit is less than 1, the series converges absolutely.
• If the limit is greater than 1, the series diverges.
• If the limit equals 1, the test is inconclusive.

For our series, the application of the ratio test reveals that the radius of convergence is 0. This outcome signals that for almost all values of \( x \), the series does not converge, except perhaps at the point of centering.

Interval of Convergence

The interval of convergence is the set of all \( x \) values for which the series converges. Once we determine the radius of convergence, it helps in identifying this interval.

• If the radius is \( R \), the interval of convergence is generally \((c-R, c+R) \) where \( c \) is the center.
• The ends of the interval need further checking for convergence separately.

In our specific problem, since the radius of convergence is zero, the series only converges at a single point: \( x=4 \). Hence, the interval is just \( x = 4 \), a rather peculiar and rare case for a power series.

Absolute Convergence

Absolute convergence of a series means that \( \sum |a_n| \) converges. This kind of convergence is strong; if a series converges absolutely, it also converges in the regular sense.

• Absolute convergence often indicates stability under rearrangements of terms.
• For many series, testing for absolute convergence involves the ratio test or the comparison test.

In our case, since the interval of convergence is one point \( x=4 \), absolute convergence simplifies here. At \( x = 4 \), the series becomes 0, affirming that it converges absolutely to that trivial value.

Conditional Convergence

Conditional convergence occurs when a series converges, but does not converge absolutely. This is a more delicate type of convergence which can change if series terms are rearranged.

• A conditionally convergent series can exhibit paradoxical results.
• This generally requires the absence of absolute convergence over some interval.

In our exercise, the series only converges trivially at \( x=4 \). Given this peculiarity, the series doesn't meet conditions that lead to classic conditional convergence over a range.