Problem 71 - Thomas Calculus

Question

It is not yet known whether the series \begin{equation}\sum_{n=1}^{\infty} \frac{1}{n^{3} \sin ^{2} n}\end{equation} converges or diverges. Use a CAS to explore the behavior of the series by performing the following steps. \begin{equation} \begin{array}{l}{\text { a. Define the sequence of partial sums }}\end{array} \end{equation} \begin{equation} s_{k}=\sum_{n=1}^{k} \frac{1}{n^{3} \sin ^{2} n}. \end{equation} \begin{equation} \begin{array}{l}{\text { What happens when you try to find the limit of } s_{k} \text { as } k \rightarrow \infty ?} \\ {\text { Does your CAS find a closed form answer for this limit? }}\end{array} \end{equation} \begin{equation} \begin{array}{l}{\text { b. Plot the first } 100 \text { points }\left(k, s_{k}\right) \text { for the sequence of partial }} \\ \quad {\text { sums. Do they appear to converge? What would you estimate }} \\ \quad {\text { the limit to be? }} \\ {\text { c. Next plot the first } 200 \text { points }\left(k, s_{k}\right) . \text { Discuss the behavior in }} \\ \quad {\text { your own words. }} \\ {\text { d. Plot the first } 400 \text { points }\left(k, s_{k}\right) \text { . What happens when } k=355 \text { ? }} \\\ \quad {\text { Calculate the number } 355 / 113 . \text { Explain from you calculation }} \\ \quad {\text { what happened at } k=355 . \text { For what values of } k \text { would you }} \\ \quad {\text { guess this behavior might occur again? }} \end{array} \end{equation}

Step-by-Step Solution

Answer

The series likely diverges due to the undefined behavior at specific points where \( \sin n \approaches 0 \), such as when \( n \approx m\pi \).

Step 1: Understanding Partial Sums

First, we define the sequence of partial sums, denoted as \( s_k = \sum_{n=1}^{k} \frac{1}{n^{3} \sin^2 n} \). The goal is to investigate the limit of \( s_k \) as \( k \rightarrow \infty \).

Step 2: Evaluate Limit with CAS

Using a Computer Algebra System (CAS), we attempt to find the limit of \( s_k \) as \( k \rightarrow \infty \). The CAS does not find a closed-form solution, suggesting that the series may diverge.

Step 3: Plot First 100 Points

Plot the first 100 points \((k, s_k)\). Observe the trend to estimate the limit. The plot may show fluctuations due to \( \sin^2 n \), but it is unlikely to show clear convergence.

Step 4: Plot First 200 Points

Plot the first 200 points \((k, s_k)\) to analyze the behavior further. The sequence continues to show oscillations, likely not stabilizing to a specific value, hinting at divergence.

Step 5: Plot First 400 Points

Plot the first 400 points \((k, s_k)\). Notice the behavior at \(k=355\). Calculate \( 355/113 \approx 3.14159... \), close to \( \pi \). This indicates that \( \sin(355) \approx 0 \), greatly increasing \( s_k \) due to \( 1/\sin^2 n \approx 1/0^2 \).

Step 6: Consider Specific k Values

Values of \( k \) that are close to multiples of \( \pi \) may cause similar behavior because \( \sin(n) \) can be near zero, creating large terms for the series. Investigate sequences where \( n \approx m\pi \).

Key Concepts

Partial Sums

When we talk about partial sums, we're looking at a new sequence created by summing the terms of another sequence from the start up to a certain number. For example, for the series given, the partial sum is represented as \( s_k = \sum_{n=1}^{k} \frac{1}{n^3 \sin^2 n} \). This means we add up the first \( k \) terms of the series.

Partial sums help us determine if a series converges or diverges. By investigating how these sums behave as \( k \) increases, we gain insight into the nature of the series. If the sequence of partial sums approaches a finite number as \( k \to \infty \), the series converges. Otherwise, it diverges. In our exercise, we explore many terms \( s_k \) to check whether they stabilize to a particular value.

However, because our series involves \( \sin^2 n \), the behavior might not be straightforward. At certain values of \( n \), especially near multiples of \( \pi \), \( \sin n \) can be very small, causing spikes in the partial sums and making it challenging to determine convergence without close analysis.

Limit of a Sequence

The limit of a sequence is a core concept in understanding series convergence. When we say we're looking for the limit of a sequence, we're analyzing what value the sequence approaches as the term number goes to infinity.

In the context of our exercise, the question is whether the partial sums \( s_k = \sum_{n=1}^k \frac{1}{n^3 \sin^2 n} \) tend to a particular number as \( k \to \infty \). If all these partial sums reach a certain number, this shows convergence. However, our series has specific complexities due to the \( \sin^2 n \) term that affects the sequence's behavior.

• A Computer Algebra System (CAS) is used to try and discover the limit. But it doesn’t find a simple solution, indicating that the series might not converge simply.
• Instead, plotting and observing have been suggested to get an intuitive sense of what the limit might be.

Plotting the sequence of partial sums for increasing terms shows how it's behaving. If sequence fluctuations do not level out but keep oscillating significantly, the series likely diverges.

Oscillating Series

An oscillating series refers to series whose terms fluctuate up and down, which can make determining convergence more complicated. In our series, because of the \( \sin^2 n \) term, the values for the term \( \frac{1}{n^3 \sin^2 n} \) can oscillate significantly depending on \( n \).

The consequence is that the partial sums don't smoothly approach a single value. One key aspect is that \( \sin n \) becomes very small for numbers close to multiples of \( \pi \), leading to large spikes in the values of \( \frac{1}{n^3 \sin^2 n} \).

• For example, as seen at \( k = 355 \), certain values make \( \sin n \) so small that it increases the partial sum dramatically. This indicates a kind of cyclical behavior based on \( n \), where at certain points, terms can grow large, showing significant oscillation.
• When plotting this series, encountering oscillations means it's dynamically jumping to larger values abruptly rather than settling down.

These properties indicate it's an oscillating series, contributing to why a CAS might struggle to find a simple limit, and why the series may not converge neatly to a single finite number.