Problem 7 - 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. $$ \sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n + 1 } \frac { 2 ^ { n } } { n ^ { 2 } } $$

Step-by-Step Solution

Answer

The series diverges.

Step 1: Identify the Alternating Series

Our given series is \( \sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n + 1 } \frac { 2 ^ { n } } { n ^ {2 } } \). This series is an alternating series because of the factor \(( - 1 ) ^ { n + 1 }\), which causes the sign of each term to alternate.

Step 2: Apply the Alternating Series Test

For an alternating series \( \sum (-1)^{n+1} a_n \) to converge, the following conditions must be satisfied: 1. \( a_n > 0 \) 2. \( a_{n+1} \leq a_n \) for all \(n\) (non-increasing)3. \( \lim_{n \to \infty} a_n = 0 \). Let's check these conditions.

Step 3: Check if \( a_n > 0 \)

The terms of our series, \( a_n = \frac{2^n}{n^2} \), are all positive since both the numerator and the denominator are positive for all \( n \geq 1 \). Thus, the first condition is satisfied.

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

Let's evaluate \( a_n = \frac{2^n}{n^2} \) for monotonicity. As \( n \) increases, the numerator \( 2^n \) grows much faster than the denominator \( n^2 \). Hence, \( a_n \) is increasing instead of non-increasing.

Step 5: Check \( \lim_{n \to \infty} a_n = 0 \)

We need to check the limit \( \lim_{n \to \infty} \frac{2^n}{n^2} \). As \( n \to \infty \), \( 2^n \) grows exponentially while \( n^2 \) grows polynomially, making the fraction approach \( \infty \), not zero. So, \( \lim_{n \to \infty} \frac{2^n}{n^2} eq 0 \).

Step 6: Conclusion

The series \( \sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n + 1 } \frac { 2 ^ { n } } { n ^ {2 } } \) does not meet the convergence criteria of the Alternating Series Test because it is not non-increasing, and its limit is not zero as \( n \to \infty \). Therefore, the series diverges.

Key Concepts

convergence and divergence

When we talk about convergence and divergence, we are discussing the behavior of sequences or series as we extend them towards infinity. In mathematical terms, convergence means that the terms of a sequence or series approach a fixed number, called a limit, as you continue to add more terms. Divergence, on the other hand, means there is no fixed point that the sequence or series approaches.
Let's explore the processes to determine if a series converges or diverges. Take, for example, our alternating series from the exercise, where the series alternates signs. You can use specific tests to judge its behavior.

• The Alternating Series Test requires that the absolute value of terms, \(a_n\), must get smaller and approach zero.
• Additionally, the terms must be non-increasing as they progress.

When these conditions are not met, as in our example where the series grows indefinitely, we conclude that the series diverges.

sequence and series

Understanding sequences and series is fundamental to addressing convergence and divergence. A sequence is simply a list of numbers following some rule. For example, in our exercise, the sequence generated by \(a_n = \frac{2^n}{n^2}\) increases at an exponential rate due to the powers of two, even though we're dividing by the square of \(n\).
A series is essentially the summing up of the sequence's terms. An alternating series, like the one in our exercise, involves altering the sign of each subsequent term. The Alternating Series Test checks certain conditions for convergence of such series, ensuring that while a sequence may continue indefinitely, the sum of its terms doesn’t explode to infinity but instead hits a steady point.
It’s vital to assess each series using its own characteristics to apply the correct convergence test.

infinite series

An infinite series is a sum of infinitely many terms. It's like opening a bottomless bag where you keep adding stuff. Mathematically, we denote this with an infinite sum symbol. Infinite series can perplex students because there isn't a clear stopping point, unlike finite series. Therefore, mathematical tests are vital in predicting their behavior.
Take the alternating series in our problem, represented as \(\sum (-1)^{n+1} \frac{2^n}{n^2}\). The core idea is to determine if by adding more terms, does it approach a finite number (converge) or grow infinitely without bounds (diverge).

• If an infinite series converges, we can calculate its sum precisely.
• If it diverges, like our example, it implies there's no single finite value we can rely on for its sum.

Understanding the nature of terms in the series, their rates of increase or decrease, and their behavior as \(n\) approaches infinity is crucial in working with infinite series.