Problem 43 - Single Variable Calculus

Question

Suppose that the interval of convergence of the series \(\sum_{n=0}^{\infty} a_{n}(x-c)^{n}\) is \((c-R, c+R]\). Prove that the series is conditionally convergent at \(c+R\).

Step-by-Step Solution

Answer

Since the series converges at \(x=c+R\) and the interval of convergence is \((c-R, c+R]\), this implies the series is not absolutely convergent at the point \(x=c-R\). Therefore, the series \(\sum_{n=0}^{\infty} a_{n}(x-c)^{n}\) is conditionally convergent at \(x=c+R\).

Step 1: Check Convergence at the Given Point

: We need to check the convergence of the following power series at the point \(x=c+R\): \[\sum_{n=0}^{\infty} a_{n}((c+R)-c)^{n}\] which simplifies to: \[\sum_{n=0}^{\infty} a_{n}R^{n}\] Since the interval of convergence is given as \((c-R, c+R]\), this means that the series converges at \(x=c+R\).

Step 2: Check Absolute Convergence

: Now we need to check if the series converges absolutely or conditionally at the point \(x=c+R\). To do this, we'll look at the absolute sum of the series: \[\sum_{n=0}^{\infty} |a_{n}R^{n}|\] We have no information about the absolute convergence of the series at the point \(x=c+R\), so we cannot conclude whether it converges or not.

Step 3: Check Conditional Convergence

: Since we know that the series converges at the point \(x=c+R\), if we can show that it does not converge absolutely at this point, then we can conclude that it converges conditionally. Recall that a series is conditionally convergent if it converges but not absolutely. Since we have already established that the series converges at \(x=c+R\), it's sufficient to show that the series does not converge absolutely at this point. We have no information about the absolute sum of the series when \(x=c+R\). However, we are given that the series converges at this point and that the interval of convergence is \((c-R, c+R]\). This implies that the series is not absolutely convergent at the point \(x=c-R\), otherwise, the interval of convergence would include that point as well. Therefore, since the series converges at \(x=c+R\) but does not converge absolutely, we can conclude that it is conditionally convergent at the point \(x=c+R\). So, the series \(\sum_{n=0}^{\infty} a_{n}(x-c)^{n}\) is conditionally convergent at \(x=c+R\).

Key Concepts

Interval of Convergence

When studying power series, the interval of convergence is a crucial concept. It refers to the set of all real numbers for which a power series converges. For the series \(\sum_{n=0}^{\infty} a_{n}(x-c)^{n}\), the interval is \((c-R, c+R]\). This means the series converges for all \(x\) in that interval.
This interval gives us valuable insight:

• Convergence is guaranteed within the open interval \((c-R, c+R)\).
• Convergence at the endpoint \(x=c+R\) is specified.

However, the behavior at the other endpoint \(x=c-R\) is typically unknown or non-convergent in this context.

Understanding the interval helps us know where the series acts like a nice, summable function.

Power Series

A power series is a type of infinite series similar to a polynomial but with infinitely many terms. It is expressed as \(\sum_{n=0}^{\infty} a_{n}(x-c)^{n}\), where \(c\) is the center of the series and \(a_n\) are coefficients. The series takes the form of powers of \(x-c\).

Why are power series important?

• They represent functions in a form that can be easily manipulated.
• They are used in calculus to approximate functions.
• Power series can be differentiated and integrated term by term within the interval of convergence.

These characteristics make power series a powerful tool in mathematics for modeling and solving problems.

Absolute Convergence

Absolute convergence means that the series of absolute values converges. For a series \(\sum_{n=0}^{\infty} a_{n}R^{n}\), if \(\sum_{n=0}^{\infty} |a_{n}R^{n}|\) also converges, we have absolute convergence.
This is a stronger form of convergence because absolute convergence implies ordinary convergence, but not vice versa.

If a series converges absolutely:

• Rearranging terms doesn't affect the sum.
• The series is stable under various mathematical operations.

For our series, we lacked sufficient information to conclude absolute convergence, leading to a focus on conditional convergence.

Convergence Tests

Convergence tests are tools we use to determine if a series converges. There are several tests available, each with different applications. When checking convergence at a point like \(x=c+R\), these tests help decide if a series converges or not.

Some common convergence tests include:

• Ratio Test: Useful for power series, especially when dealing with factorials or exponentials.
• Root Test: Applies well when each term is raised to a power.
• Comparison Test: Compares a series to a known convergent or divergent series.
• Integral Test: Helps if the series can be related to a function that is easy to integrate.

In our context, using such tests helps determine if the series only converges or also converges absolutely, guiding us toward understanding conditional convergence.