Problem 9 - Thomas Calculus

Question

In Exercises \(1 - 14 ,\) determine if the alternating series converges or diverges. Some of the series do not satisfy the conditions of the Alternating Series Test. $$ \sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n + 1 } \left( \frac { n } { 10 } \right) ^ { n } $$

Step-by-Step Solution

Answer

The series diverges.

Step 1: Write out the series formula

The given series is \( \sum_{n=1}^{\infty} (-1)^{n+1} \left( \frac{n}{10} \right)^n \). This is an alternating series because of the \((-1)^{n+1}\) factor, which causes the signs of the terms to alternate.

Step 2: Verify the Alternating Series Conditions

According to the Alternating Series Test, a series \( \sum (-1)^{n+1} b_n \) converges if: (i) \( b_n \) is positive, (ii) \( b_n \) is non-increasing, and (iii) \( \lim_{n \to \infty} b_n = 0 \). Here, \( b_n = \left( \frac{n}{10} \right)^n \).

Step 3: Check positivity

Since \( b_n = \left( \frac{n}{10} \right)^n \), and both \( n \) and \( \frac{1}{10} \) are positive, \( b_n \) is positive for all \( n \geq 1 \).

Step 4: Determine if \( b_n \) is non-increasing

Calculate the ratio \( \frac{b_{n+1}}{b_n} = \left( \frac{n+1}{10} \right)^{n+1} \div \left( \frac{n}{10} \right)^n \). Simplifying, \( \frac{b_{n+1}}{b_n} = \frac{(n+1)^n}{n^n} \cdot \frac{n+1}{10} \). As \( n \) grows, \( \frac{n+1}{n} > 1 \), thus \( b_n \) is increasing, not non-increasing.

Step 5: Evaluate the limit \( \lim_{n \to \infty} b_n \)

Evaluate the limit \( \lim_{n \to \infty} \left( \frac{n}{10} \right)^n = \infty \). Since the term goes to infinity, it does not go to zero.

Step 6: Conclusion of the Alternating Series Test

The series does not satisfy the necessary conditions of the Alternating Series Test because \( \lim_{n \to \infty} b_n eq 0 \) and \( b_n \) is increasing. Therefore, the series diverges.

Key Concepts

Convergence and Divergence

Understanding whether an infinite series converges or diverges is essential in calculus. When we talk about convergence, we mean that the series approaches a specific value as more terms are added. In contrast, divergence suggests that the series does not settle on a limit and continues to grow or oscillate indefinitely.

For an alternating series like the one given, the Alternating Series Test is often used. The test has three criteria to determine convergence:

• The sequence of terms, denoted as \( b_n \), must be positive.
• The sequence must not increase as \( n \) increases (i.e., \( b_n \) should be non-increasing).
• The limit of \( b_n \) as \( n \) approaches infinity must be zero.

If any of these conditions are not met, the series will diverge. In the case of our exercise, the sequence fails two conditions: it is increasing, and its limit does not approach zero, confirming divergence.

Infinite Series

An infinite series is a sequence of numbers added indefinitely. It can be written in the form \( \sum_{n=1}^{\infty} a_n \), where \( a_n \) represents the nth term. Infinite series are at the heart of many mathematical concepts since they can describe functions, phenomena, or numbers.

Alternating series, like the one discussed here, have terms that switch signs due to factors like \((-1)^{n+1}\). This alternating nature affects whether they converge or diverge. Specifically, the absolute value of their terms must diminish consistently and ultimately approach zero for convergence through the Alternating Series Test.

Not meeting these criteria, as in our exercise, results in divergent behavior. The series will spread further from any potential limit as more terms are added.

Sequence Limits

A sequence's limit is what the terms of the sequence approach as \( n \), the term index, goes to infinity. In simpler terms, it's a value that the terms get closer and closer to as you move along the sequence.

For an alternating series, ensuring that \( b_n \) heads to zero is critical. If \( \lim_{n \to \infty} b_n = 0 \), then one of the key criteria for convergence under the Alternating Series Test is met. However, if this limit is not zero, the series cannot converge.

In the exercise's series, \( b_n = \left( \frac{n}{10} \right)^n \), and when \( n \) becomes very large, this term grows larger instead of shrinking. Therefore, the sequence limit doesn't go to zero, indicating divergence of the series.