Problem 30 - Calculus: Early Transcendentals

Question

Suppose that \( \sum_{n = 0}^{\infty} c_nx^n \) converges when \( x = - 4 \) and diverges when \( x = 6. \) What can be said about the convergence or divergence of the following series? (a) \( \sum_{n = 0}^{\infty} c_n \) (b) \( \sum_{n = 0}^{\infty} c_n8^n \) (c) \( \sum_{n = 0}^{\infty} c_n( - 3)^n \) (d) \( \sum_{n = 0}^{\infty} ( - 1)^n c_n 9^n \)

Step-by-Step Solution

Answer

(a) Converges; (b) Diverges; (c) Converges; (d) Diverges.

Step 1: Identify the Radius of Convergence

Since the series \( \sum_{n = 0}^{\infty} c_nx^n \) converges at \( x = -4 \) and diverges at \( x = 6 \), we can determine the radius of convergence \( R \) of the power series. The point \( x = -4 \) is inside the interval of convergence, while \( x = 6 \) is outside it. Therefore, the distance from the center of the interval (usually 0) to -4 gives a condition that \( |x| = 4 \) is within the interval. The divergence at \( x=6 \) means that \( |x|=6 \) is outside the interval, hence \( R < 6 \) and \( R \geq 4 \). This gives the radius of convergence as \( R = 5 \), as it must encompass 4 but not reach 6.

Step 2: Consider the Series at Different Points

Using the radius of convergence \( R = 5 \), we analyze each given series to check if \( |x| \) falls within this radius or not, for convergence.(a) For \( \sum_{n = 0}^{\infty} c_n \), \( x = 1 \), so \( |x| = 1 < 5 \). Thus, the series converges.(b) For \( \sum_{n = 0}^{\infty} c_n8^n \), \( x = 8 \), so \( |x| = 8 > 5 \). Therefore, the series diverges.(c) For \( \sum_{n = 0}^{\infty} c_n(-3)^n \), \( x = -3 \), so \( |x| = 3 < 5 \). Hence, the series converges.(d) For \( \sum_{n = 0}^{\infty} (-1)^n c_n 9^n \), \( x = 9 \), so \( |x| = 9 > 5 \). Thus, the series diverges.

Key Concepts

Power Series

A power series is a type of infinite series that is represented as a sum of terms made up of a coefficient and a variable raised to a power. It is usually expressed in the form:

• \( \, \sum_{n=0}^{\infty} c_n(x-a)^n \)

Here, \( c_n \) represents the coefficients, \( x \) is the variable, and \( a \) is the center of the series.
Power series play a crucial role because they can be used to represent functions in a broad variety of fields, such as in calculus or complex analysis.
A key aspect of understanding power series is determining the range of \( x \) values where the series converges. This range is essential for ensuring the series accurately represents a function within a certain domain.

Convergence

Convergence in the context of power series happens when the sum of the terms approaches a specific finite value as the number of terms increases.
When we say a series converges, it implies that as you add more and more terms, the series approaches a limit, rather than continually increasing or decreasing.
For a power series \( \sum_{n=0}^{\infty} c_n(x-a)^n \) to converge at a specific \( x \), the absolute value of \( x-a \) must be less than the radius of convergence \( R \). Three useful tests to determine convergence include:

• Ratio Test: This test can help identify the range of \( x \) that makes the series converge.
• Root Test: Similar to the ratio test, it provides another approach to check series convergence.
• Direct Substitution: Plugging specific values of \( x \) into the series to see if the series converges.

Understanding these basic tests helps in identifying the intervals where power series remain valid.

Divergence

When a series does not converge, it diverges. This means that as more and more terms are added, the series does not approach a finite limit but instead increases indefinitely or oscillates.
For a power series \( \sum_{n=0}^{\infty} c_n(x-a)^n \), divergence occurs when the term \( c_n(x-a)^n \) causes the series sum to grow without bound or not settle to a specific value.In terms of radius:

• If the absolute value of \( x-a \) is greater than the radius of convergence \( R \), the series will diverge.
• If \( x-a \) is exactly equal to \( R \), the behavior of the series must be checked case by case.

It is useful to examine the endpoints separately, since sometimes series can behave differently at these points than within the interval. Understanding divergence helps in determining when a power series is inappropriate for representing a function at certain points.

Interval of Convergence

The interval of convergence is the range of \( x \) values for which a power series converges. It is crucial because it tells us where a power series is valid for representing a function.
For a power series centered at \( a \), the interval is generally written as:

• \( (a-R, a+R) \)

Here, \( R \) is the radius of convergence, and the interval includes all \( x \) values where \( |x-a| < R \).Endpoints of the interval, \( x = a-R \) and \( x = a+R \), need to be checked individually:

• At these points, the series may converge or diverge, depending on the specific series.
• Techniques such as substitution or specific tests (e.g., Ratio Test) can determine behavior at endpoints.

Once the interval and endpoints are analyzed, one can fully understand where the series represents the function accurately.