Question

Consider the alternating series $$ \sum_{k=1}^{\infty}(-1)^{k+1} a_{k}, \text { where } a_{k}=\left\\{\begin{array}{cl} \frac{4}{k+1}, & \text { if } k \text { is odd } \\ \frac{2}{k}, & \text { if } k \text { is even } \end{array}\right. $$ a. Write out the first ten terms of the series, group them in pairs, and show that the even partial sums of the series form the (divergent) harmonic series. b. Show that \(\lim _{k \rightarrow \infty} a_{k}=0\) c. Explain why the series diverges even though the terms of the series approach zero.

Short answer

To summarize: a) The first ten terms of the alternating series are: $$2 - 1 + 1 - \frac{1}{2} + \frac{2}{3} - \frac{1}{3} + \frac{1}{2} - \frac{1}{4} + \frac{2}{5} - \frac{1}{5}$$ And the even partial sums form a divergent harmonic series: $$1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5}$$ b) We showed that the limit of \(a_k\) as \(k\) approaches infinity is zero: $$\lim_{k \rightarrow \infty} a_k = 0$$ c) The alternating series diverges because it does not meet both conditions of the Alternating Series Test. Although the terms approach zero, the series does not have strictly decreasing terms. Thus, it diverges even though the terms approach zero.

Step-by-step solution

a. First ten terms of the series and even partial sums#

To find the first ten terms of the alternating series, we will substitute the values of \(k=1,2,3,...,10\) into the expression for \(a_k\), keeping in mind the different expressions for odd and even values of \(k\). Then we'll evaluate the terms with their signs. 1. For \(k=1\) (odd): \(a_1 = \frac{4}{1+1} = 2\) and the term is positive. 2. For \(k=2\) (even): \(a_2 = \frac{2}{2} = 1\) and the term is negative. 3. For \(k=3\) (odd): \(a_3 = \frac{4}{3+1} = 1\) and the term is positive. 4. For \(k=4\) (even): \(a_4 = \frac{2}{4} = \frac{1}{2}\) and the term is negative. 5. For \(k=5\) (odd): \(a_5 = \frac{4}{5+1} = \frac{2}{3}\) and the term is positive. 6. For \(k=6\) (even): \(a_6 = \frac{2}{6} = \frac{1}{3}\) and the term is negative. 7. For \(k=7\) (odd): \(a_7 = \frac{4}{7+1} = \frac{1}{2}\) and the term is positive. 8. For \(k=8\) (even): \(a_8 = \frac{2}{8} = \frac{1}{4}\) and the term is negative. 9. For \(k=9\) (odd): \(a_9 = \frac{4}{9+1} = \frac{2}{5}\) and the term is positive. 10. For \(k=10\) (even): \(a_{10} = \frac{2}{10} = \frac{1}{5}\) and the term is negative. So the first ten terms of the series are: $$2 - 1 + 1 - \frac{1}{2} + \frac{2}{3} - \frac{1}{3} + \frac{1}{2} - \frac{1}{4} + \frac{2}{5} - \frac{1}{5}$$ Now we'll group them in pairs: $$(2 - 1) + (1 - \frac{1}{2}) + (\frac{2}{3} - \frac{1}{3}) + (\frac{1}{2} - \frac{1}{4}) + (\frac{2}{5} - \frac{1}{5})$$ Simplifying and summing the pairs, we get the even partial sums: $$1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5}$$ These sums form the harmonic series which is known to be divergent.

b. Limit of \(a_k\) as \(k\) approaches infinity#

To show that \(\lim _{k \rightarrow \infty} a_{k}=0\), we need to examine the limit of the expressions for \(a_k\) as \(k\to\infty\) separately for odd and even values of \(k\). For odd \(k\): $$\lim _{k \rightarrow \infty, k \text{ odd}} \frac{4}{k+1} = \frac{4}{\infty} = 0 $$ For even \(k\): $$\lim _{k \rightarrow \infty, k \text{ even}} \frac{2}{k} = \frac{2}{\infty} = 0 $$ Since the limit is zero for both odd and even \(k\), we can conclude that \(\lim_{k \rightarrow \infty} a_k = 0\).

c. Divergence of the series despite terms approaching zero#

The alternating series diverges even though its terms approach zero because the two conditions of the Alternating Series Test are not met. The Alternating Series Test states that a series converges if: 1. The terms \(a_k\) are decreasing: \(a_{k+1} \le a_k\) for all \(k\). 2. The limit of \(a_k\) as \(k \rightarrow \infty\) is zero: \(\lim_{k \rightarrow \infty} a_k = 0\) We have already shown in part b that the limit of \(a_k\) approaches zero, so the second condition is met. However, the first condition is not met in this case. Examining the terms of the series, we see that the terms do not strictly decrease as \(k\) increases. For example, \(a_2=1\) while \(a_3=1\), and \(a_7=\frac{1}{2}\) while \(a_4=\frac{1}{2}\). As the series does not meet both conditions of the Alternating Series Test, it diverges even though the terms approach zero.

Key concepts

Divergence of Series

The concept of divergence in series is a fundamental topic in calculus. A series is said to diverge if the sequence of its partial sums does not converge to a finite limit. In simpler terms, a divergent series doesn't settle down to a specific number as you keep adding more and more terms.

Continuing to add terms forever can result in a sum that grows without bound or oscillates indefinitely. The presence of divergence, especially in alternating series, can seem counterintuitive because sometimes the terms get smaller over time. However, as shown in the exercise, even when terms approach zero, divergence can still occur. This happens when the necessary conditions for convergence aren't fully met.

In the Alternating Series Test, one condition is that the absolute values of the terms must decrease monotonically. If the series' terms don't follow a decreasing pattern in magnitude, then they might fail the test and cause divergence. As explained in our example, although terms approach zero by getting infinitely smaller, the inconsistent pattern of decreasing magnitude led to divergence. Hence, careful analysis of both conditions and term behaviors is crucial in understanding series divergence.

Harmonic Series

The harmonic series is an important concept to grasp, as it is a classic example of a divergent series. The harmonic series is defined as the sum of reciprocals of positive integers. Mathematically, this can be expressed as:

• \( ext{Harmonic Series} = 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \cdots \).

The harmonic series illustrates divergence even though each term individually becomes very small as more terms are added. Despite this, the series does not approach a limit.

Interestingly, while each term approaches zero, the sum grows indefinitely. This might be surprising because individually tiny terms add up to infinity when dealing with infinitely many terms. In our exercise, by grouping and summing the series' terms in pairs, we demonstrated how the partial sums mimic this characteristic and form a divergent harmonic sequence.

Understanding the harmonic series is critical, as it provides a stepping stone to recognizing patterns in more complex problems. Its divergence exemplifies how an infinite series can behave non-convergently despite initial expectations.

Limits in Calculus

Limits are foundational in calculus and crucial for analyzing the behavior of sequences and series. A limit tells us the value that a function or sequence "approaches" as the input (or index) reaches some point, often infinity. In the context of the given exercise, we're particularly interested in the limits of the series' terms as \(k\) approaches infinity.

Showing that \( \lim_{k \to \infty} a_k = 0\) is a key step in evaluating series. This confirms that each term individually diminishes to zero. However, it's essential to remember that while reaching the limit of zero satisfies one condition for convergence in alternating series, it doesn't guarantee the entire series will converge.

The limit allows us to assess the long-term behavior of these terms and helps determine the overall properties of the series, such as convergence or divergence. Limits serve as a mathematical microscope, showing us the fate of infinite sequences and whether a series might settle to a finite sum or diverge indefinitely.

Comprehending limits is thus not just an academic exercise but a necessary skill for deeper analysis in calculus.