Problem 2 - 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? $$\sum_{n=0}^{\infty}(x+5)^{n}$$

Step-by-Step Solution

Answer

The radius of convergence is 1, with the interval \((-6, -4)\). The series converges absolutely there, and conditionally nowhere.

Step 1: Identifying the Series Type

The given series \( \sum_{n=0}^{\infty} (x+5)^n \) is a geometric series with a common term \( a_n = (x+5)^n \). Geometric series are characterized by their common ratio.

Step 2: Determine Common Ratio

For a geometric series \( \sum_{n=0}^{\infty} ar^n \), the common ratio is \( r \). For our series, the common ratio is \( r = x+5 \).

Step 3: Finding the Radius of Convergence

A geometric series converges if \( |r| < 1 \). Therefore, we need \(|x+5| < 1\). This gives two inequalities: \( -1 < x+5 < 1 \).

Step 4: Solving the Inequality for Interval of Convergence

Solving \(-1 < x+5 < 1\) gives \(-6 < x < -4\). This is the interval of convergence for the series where the series converges absolutely.

Step 5: Check Absolute and Conditional Convergence

For geometric series, it converges absolutely wherever it converges, since no extra factors are present. Since there are no additional terms or factorials, there is no conditional convergence region separate from the absolute convergence.

Key Concepts

Geometric Series

A geometric series is a type of infinite series where every term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. This series is represented as \( \sum_{n=0}^{\infty} ar^n \), where \( a \) is the initial term, and \( r \) is the common ratio. In our particular exercise, the series can be expressed as \( \sum_{n=0}^{\infty} (x+5)^n \). This tells us:

• The first term \( a = 1 \), and
• The common ratio \( r = x+5 \).

Geometric series have fascinating properties because whether they converge or not depends entirely on the value of \( r \). Converging in this context means that as we add more terms, the sum approaches a finite value.

Interval of Convergence

The interval of convergence refers to the set of all \( x \) values for which an infinite series achieves convergence. For a geometric series, this convergence criterion is met when the absolute value of the common ratio is less than 1, i.e., \( |r| < 1 \).

For our series, this requirement becomes \( |x+5| < 1 \). To solve this inequality, we break it down into:

• \( x+5 > -1 \) resulting in \( x > -6 \)
• \( x+5 < 1 \) giving us \( x < -4 \).

Putting these together, we get the interval of convergence as \( -6 < x < -4 \). Within this specific range, the series will converge and generate a specific finite sum.

Absolute Convergence

Absolute convergence of a series means that the series \( \sum_{n=0}^{\infty} a_n \) converges even when we replace all terms with their absolute values, \( \sum_{n=0}^{\infty} |a_n| \).

In a geometric series like \( \sum_{n=0}^{\infty} (x+5)^n \), wherever the series converges, it necessarily does so absolutely due to its structure: no additional terms alter the behavior other than the basic multiplication factor.

For this exercise, the geometric series converges absolutely within the interval \( -6 < x < -4 \), driven by the condition \( |x+5| < 1 \). Absolute convergence tends to simplify analysis because peculiar oscillations or conditional terms don't interfere, allowing for cleaner calculations.

Conditional Convergence

Conditional convergence occurs when a series converges due to the interaction of its terms, but diverges when all terms are replaced with their absolute values.

While this phenomenon can appear in other types of series, a geometric series like \( \sum_{n=0}^{\infty} (x+5)^n \) doesn’t exhibit conditional convergence. Here,

• The convergence is absolute across the interval \( -6 < x < -4 \),
• And there are no adjustments needed to determine the nature of convergence in distinct regions.

Thus, if a geometric series converges, it will do so absolutely and does not split into separate regions for absolute and conditional convergence as one might find in more complex series.