Problem 1 - 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? $$\sum_{n=0}^{\infty} x^{n}$$

Step-by-Step Solution

Answer

(a) Radius 1, interval \\(-1 < x < 1\\); (b) Absolutely for \\(-1 < x < 1\\); (c) No conditional convergence.

Step 1: Identify the Series

The given series is a geometric series \(\sum_{n=0}^{\infty} x^{n}\). A geometric series converges if the absolute value of the common ratio \(|r| < 1|\). Here, \(r = x\).

Step 2: Determine the Radius of Convergence

For the geometric series \(\sum_{n=0}^{\infty} x^{n}\), the radius of convergence \(R\) is determined by \(|x| < 1\). Thus, \(R = 1\).

Step 3: Find the Interval of Convergence

The interval of convergence is determined by solving \( |x| < 1 \), which results in \(-1 < x < 1\).

Step 4: Check for Absolute Convergence

The series \(\sum_{n=0}^{\infty} x^{n}\) converges absolutely if it converges in the interval \(-1 < x < 1\). Since it converges only when \(|x| < 1\), this is also the region for absolute convergence.

Step 5: Evaluate Conditional Convergence

For a geometric series, conditional convergence is not applicable because it either converges absolutely or diverges outside the radius of convergence. Therefore, the series does not converge conditionally anywhere.

Key Concepts

Geometric Series

A geometric series is a simple type of series where each term is a constant multiple, known as the common ratio, of the previous term. A classic example of this is the series of the form \( \sum_{n=0}^{\infty} ar^n \), where \( a \) is the initial term, and \( r \) is the common ratio. This series will converge—that is, approach a finite value—if the absolute value of the common ratio, \( |r| \), is less than 1.

• The series \( \sum_{n=0}^{\infty} x^n \) is an example of a geometric series with \( a = 1 \) and \( r = x \).
• If \( |x| < 1 \), the series converges to \( \frac{1}{1-x} \).

Remember that outside this range, the series diverges. The simple structure of geometric series makes them easy to analyze, especially when calculating the radius of convergence.

Interval of Convergence

The interval of convergence is a crucial concept that identifies values of \( x \) for which a series converges. In the context of our geometric series \( \sum_{n=0}^{\infty} x^n \), determining this involves finding for what \( x \) the series remains finite.

To find this interval:

• We solve \( |x| < 1 \), yielding the interval \(-1 < x < 1\).

This interval tells us precisely where the series converges. Importantly, for most geometric series, the convergence is straightforward because it only occurs in this clearly defined range. Also, note that geometric series have no conditional convergence range, as they either converge absolutely within their interval or diverge outside of it.

Absolute Convergence

Absolute convergence refers to when the series of absolute values itself converges. For our geometric series \( \sum_{n=0}^{\infty} x^n \), determining absolute convergence is simple because:

• The convergence depends solely on whether \( |x| < 1 \).

Within the interval \(-1 < x < 1\), the geometric series converges absolutely. This means that if you replace all terms by their absolute values, the new series will still converge. Absolute convergence is a robust form of convergence, ensuring that the rearrangement of terms does not affect the sum of the series.

Conditional Convergence

Conditional convergence occurs if a series converges, but the series of absolute values does not converge. This is a characteristic not found in geometric series.

For the geometric series \( \sum_{n=0}^{\infty} x^n \), we will never find conditional convergence. Here's why:

• It either converges absolutely within the interval \(-1 < x < 1\), or it diverges completely outside of that interval.

This is because geometric series are quite predictable: they only exhibit absolute convergence or no convergence at all. Therefore, when dealing with geometric series, you can focus solely on absolute convergence and follow the simple rule of thumb that \( |x| < 1 \).