Question

The following series do not satisfy the hypotheses of the alternating series test as stated. In each case, state which hypothesis is not satisfied. State whether the series converges absolutely.Show that the alternating series \(\frac{2}{3}-\frac{3}{5}+\frac{4}{7}-\frac{5}{9}+\cdots\) does not converge. What hypothesis of the alternating series test is not met?

Short answer

The series diverges due to non-decreasing terms; it fails the alternating series test.

Step-by-step solution

Identify the General Term

First, identify the general term of the given alternating series. The series can be written as: \[ a_n = \frac{n+1}{2n+1} (-1)^{n+1} \] where \( n \) starts from 1. The series is of the form: \[ \sum_{n=1}^{\infty} (\frac{n+1}{2n+1} (-1)^{n+1}) \]

Check Decreasing Behavior

We need to verify if the absolute values of the terms decrease, which is the second condition in the alternating series test. Compare \( a_n \) and \( a_{n+1} \): \[ |a_n| = \frac{n+1}{2n+1}, |a_{n+1}| = \frac{n+2}{2n+3} \] To check if \( |a_{n+1}| < |a_n| \) we need to compare: \[ \frac{n+2}{2n+3} < \frac{n+1}{2n+1} \] Cross-multiply and simplify: \[ (n+2)(2n+1) < (n+1)(2n+3) \] Upon simplification: \[ 2n^2 + 5n + 2 < 2n^2 + 5n + 3 \] which is not true for all \( n \). Thus, the series terms are not decreasing.

Determine Absolute Convergence

Examine the absolute convergence by considering the non-alternating series:\[ \sum_{n=1}^{\infty} \frac{n+1}{2n+1} \]We need to evaluate this series. Notice that the dominant behaviors in the numerator and the denominator are \( n \), so:\[ \lim_{n \to \infty} \frac{n+1}{2n+1} = \frac{1}{2} \]Since the terms do not approach zero, the series does not converge absolutely.

Check the Alternating Series Test Hypotheses

The alternating series test has two main criteria: the terms \( a_n \) must eventually decrease in absolute value, and \( \lim_{n \to \infty} a_n = 0 \). Here, the series fails the first criterion as shown in 'Check Decreasing Behavior' since \( |a_n| \) does not form a decreasing sequence.

Conclusion on Series Convergence

Since neither the alternating series test nor the absolute convergence test confirms convergence, the series diverges. The absence of satisfying the decreasing behavior criterion is the primary reason the series fails the alternating series test.

Key concepts

Series Convergence

In mathematics, when we discuss series convergence, we are essentially examining whether adding up the infinite terms of a series results in a finite sum. Understanding this concept is crucial when analyzing whether an alternating series, such as the one presented in the problem, converges.Alternating series are composed of terms that switch sign, usually involving a factor of \((-1)^n\) or similar. Convergence in these series isn't guaranteed simply by the alternating nature. Instead, specific tests, like the alternating series test, help us determine if they sum to a finite number. However, we must thoroughly check if the series satisfies all the test criteria. If even one criterion isn't met, as seen in the original exercise, the series doesn't converge.To recap, ensuring that a series converges involves checking criteria such as whether the terms eventually decrease in absolute value and if the sequence approaches zero. Without these assurances, the series will diverge, meaning the sum stretches towards infinity or fails to land on a finite number.

Absolute Convergence

Absolute convergence is a stronger form of convergence for infinite series, where instead of just considering the series, we examine the sum of the absolute values of its terms. If a series converges absolutely, its positive-only counterpart also converges to a finite value.Why is this important? An absolutely convergent series guarantees convergence without doubt, even if the series alternates or has varying signs. In the context of the exercise given, examining the absolute convergence involves removing alternating signs and assessing if the resulting series converges. The given series \(\sum_{n=1}^{\infty} \frac{n+1}{2n+1}(-1)^{n+1}\) fails absolute convergence because replacing each term with its absolute value results in a series whose limit is non-zero:\(\lim_{n \to \infty} \frac{n+1}{2n+1} = \frac{1}{2}\).When already a series doesn't tend towards zero, we must conclude that absolute convergence is absent. This failure often indicates the series itself also diverges.

Decreasing Sequence

A decreasing sequence plays a crucial role in the alternating series test, which helps determine convergence. For the alternating series test to be applicable, the absolute values of the terms in the series must form a decreasing sequence. This means each subsequent term must be smaller than the one preceding it in absolute value.The problem in the exercise specifically highlights the necessity of verifying the decreasing nature of the sequence. When comparing the terms \(|a_n| = \frac{n+1}{2n+1}\) with \(|a_{n+1}| = \frac{n+2}{2n+3}\), the simplification reveals that the inequality \(\frac{n+2}{2n+3} < \frac{n+1}{2n+1}\) does not hold for all \(n\). This suggests that the sequence fails to be consistently decreasing.Without this property, the hypothesis for the alternating series cannot be satisfied. Consequently, the alternating series test fails, indicating another need to further examine convergence through absolute convergence or other tests.