Question

Show that the series $$\frac{1}{3}-\frac{2}{5}+\frac{3}{7}-\frac{4}{9}+\cdots=\sum_{k=1}^{\infty}(-1)^{k+1} \frac{k}{2 k+1}$$ diverges. Which condition of the Alternating Series Test is not satisfied?

Short answer

Answer: The second condition, $$\lim_{n \to \infty} a_n = 0$$, is not satisfied by the series.

Step-by-step solution

Write down the general term a_n

The series can be written in the general form: $$\sum_{k=1}^{\infty}(-1)^{k+1}\frac{k}{2 k+1}$$. So, the general term a_n will be: $$a_n = \frac{n}{2n+1}$$

Check if the terms are non-increasing

To check if the terms are non-increasing, we will examine whether $$a_{n+1} \leq a_n$$ or not. Let's compare the terms: $$a_{n+1} = \frac{n+1}{2(n+1)+1}, \quad a_n = \frac{n}{2n+1}$$ Now, we can check if $$\frac{n+1}{2n+3} \leq \frac{n}{2n+1}$$ After cross-multiplying, we get: $$n(2n+3) \leq (n+1)(2n+1)$$. Simplifying, we have: $$2n^2 + 3n \leq 2n^2 + 3n + 2$$. Therefore, we see that $$a_{n+1} \leq a_n$$, and thus the first condition of the AST is satisfied.

Check if the limit of the terms is zero

To check if the limit of the terms approaches zero, we will find the limit of the general term a_n as n approaches infinity: $$\lim_{n \to \infty} \frac{n}{2n+1}$$. We can divide the numerator and denominator by n, which results in: $$\lim_{n \to \infty} \frac{1}{2+\frac{1}{n}}$$. Now, as n approaches infinity, $$\frac{1}{n}$$ approaches 0, so we have: $$\lim_{n \to \infty} \frac{1}{2+0} = \frac{1}{2}$$. Since the limit is not zero, the second condition of the AST is not satisfied. As the second condition of the AST is not satisfied, we can conclude that the series $$\sum_{k=1}^{\infty}(-1)^{k+1}\frac{k}{2 k+1}$$ diverges. The condition which is not satisfied is:$$\lim_{n \to \infty} a_n = 0$$.

Key concepts

Series Convergence

Understanding series convergence is vital for determining the behavior of an infinite series. When a series converges, the sum of its terms approaches a specific finite value as more terms are added. This concept is essential for predicting the behavior and value of series in calculus. For an alternating series like the one in the original exercise, the Alternating Series Test (AST) helps verify convergence. This test checks two main conditions:

• The absolute value of the terms must be non-increasing, meaning each term is no larger than the previous term.
• The limit of the terms as they approach infinity must be zero.

In the original solution, we verified that the first condition is satisfied, as each term was shown to be non-increasing. However, it is the second condition that fails, leading us to investigate the result of divergence rather than convergence.

Divergence

Divergence in series analyses is the conclusion that the sum does not settle at a particular value, even as the number of terms grows towards infinity. When a series diverges, no finite sum can be determined, which is crucial for mathematical predictions and calculations. For the series in question, \(rac{1}{3}-rac{2}{5}+rac{3}{7}-rac{4}{9}+ ext{and so on.}\)The Alternating Series Test reveals divergence because the series does not meet all necessary conditions for convergence. Specifically, while the terms of the series are non-increasing, the limit of the term does not equal zero. This blockage is the crucial aspect where the series fails to converge, confirming its divergence. Understanding why a series diverges is as important as understanding convergence, as it can influence broader mathematical models and calculations.

Limits

The concept of limits plays a fundamental role in determining the behavior of functions and series in calculus. When examining an infinite series, one of the key questions concerns the limit of its terms. A limit answers what value the terms of a sequence or series approach as they extend indefinitely. In the Alternating Series Test context, the limit is used to determine if a crucial condition is satisfied: \( ext{lim}_{n \to \infty} a_n = 0.\)For the series under consideration, the limit calculation involved evaluating \( ext{lim}_{n \to \infty} \frac{n}{2n+1},\)which simplifies to \(\frac{1}{2},\)Not zero, as required for convergence. This non-zero limit is the stumbling block that results in divergence. Limits are pivotal in countless areas of mathematics beyond series analysis, providing foundational insights into how systems behave as parameters grow indefinitely.