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 series converges absolutely for \(-6 < x < -4\); no conditional convergence.

Step 1: Identify the Series and General Form

The given series is \( \sum_{n=0}^{\infty} (x+5)^n \). This is a geometric series with the general form \( a_n = (x+5)^n \) where the first term \( a = 1 \) and the common ratio \( r = x+5 \). For a geometric series, we know that it converges if the absolute value of the common ratio is less than 1, i.e., \( |r| < 1 \). In this case, \( |x+5| < 1 \).

Step 2: Solve for the Interval of Convergence

To find the interval of convergence, solve the inequality \( |x+5| < 1 \).This translates to \(-1 < x+5 < 1\).Subtract 5 from all sides of the inequality:\[-1 - 5 < x + 5 - 5 < 1 - 5\]Simplifying gives \(-6 < x < -4\). Thus, the interval of convergence is \(-6, -4\).

Step 3: Determine Absolute Convergence

A geometric series either converges or diverges absolutely based on the value of \( r \). Since the series is geometric, it does not have conditions for absolute convergence separately as it's already covered by where it converges. If \( -6 < x < -4 \), the series converges absolutely.

Step 4: Determine Conditional Convergence

For geometric series like \( \sum_{n=0}^{\infty} (x+5)^n \), there is no conditional convergence. It is either absolutely convergent or divergent based on the common ratio. Thus, there are no values of \( x \) where the series converges conditionally.

Key Concepts

Geometric Series

A geometric series is a sum of terms in which each term is a fixed multiple of the previous one. In other words, it is a series where each term after the first is found by multiplying the previous term by a constant called the common ratio. For example, in the series

• \( a, ar, ar^2, ar^3, \ldots \)

the first term is \( a \) and the common ratio is \( r \). The sum of an infinite geometric series can be determined if the common ratio \( |r| < 1 \). In this case, the series converges, and we can find its sum using the formula \[S = \frac{a}{1 - r}\]This formula works because as the series progresses, each additional term becomes smaller and smaller, allowing the series to approach a finite limit. When the given series is \[\sum_{n=0}^{\infty}(x+5)^n\]it takes the form of a geometric series with \( a = 1 \) and \( r = x+5 \). Therefore, to find where the series converges, we ensure that the absolute value of the common ratio is less than one, \(|x+5| < 1\).This key characteristic of geometric series is what makes them relatively simple to work with, particularly when compared to other types of series where determining convergence might require more sophisticated tests.

Interval of Convergence

The interval of convergence is the range of

• x-values

for which a series converges. For geometric series, finding this interval involves solving an inequality related to the common ratio. As we determined, the series converges where \[|x+5| < 1\]Let's solve this inequality to find the specific interval:

• First, write \( -1 < x+5 < 1 \).
• Subtract 5 from each part:\(-6 < x < -4\).

Therefore, the interval of convergence is \((-6, -4)\). This means that for any value of \( x \) within this range, the series will converge. It is essential to understand that outside this range, the series diverges and does not sum to a finite number. The interval of convergence is different for different series, and depending on whether endpoints are included, convergence can be checked separately at those points.

Absolute Convergence

Absolute convergence is a strong form of convergence for series. A series

• converges absolutely

if the series of absolute values of its terms converges. For the series \[\sum_{n=0}^{\infty} (x+5)^n\]since it is a geometric series, once we determine that \[|x+5| < 1\]ensures convergence, this is both absolute and complete for any \( x \) within the interval of convergence, which we've found to be \((-6, -4)\).Geometric series, like this one, can't exhibit conditional convergence, which occurs when a series converges but doesn't converge absolutely. This is more typical in other types of series, notably alternating ones. Thus, for this series, whenever it converges, it does so absolutely within its interval of convergence. Understanding absolute convergence helps confirm the stability of the sum of a series and is critical when analyzing or working through various kinds of series.