Question

Choose your test Use the test of your choice to determine whether the following series converge. $$\sum_{k=1}^{\infty} \frac{\sin ^{2} k}{k^{2}}$$

Short answer

Answer: The series $$\sum_{k=1}^{\infty} \frac{\sin ^{2} k}{k^{2}}$$ converges.

Step-by-step solution

Identify the comparison series

We want to find a series that we can use to compare with the given series. In this case, we can compare the series $$\sum_{k=1}^{\infty} \frac{\sin ^{2} k}{k^{2}}$$ with the series $$\sum_{k=1}^{\infty} \frac{1}{k^{2}}$$ since we know that \(0 \leq \sin^2 k \leq 1\) for all \(k \in \mathbb{N}\).

Apply the Comparison Test

The Comparison Test states that if \(0 \leq a_k \leq b_k\) for all \(k\) and the series \(\sum_{k=1}^{\infty} b_k\) converges, then the series \(\sum_{k=1}^{\infty} a_k\) also converges. In this case, we have \(0 \leq \frac{\sin ^{2} k}{k^{2}} \leq \frac{1}{k^{2}}\) with \(a_k = \frac{\sin ^{2} k}{k^{2}}\) and \(b_k = \frac{1}{k^{2}}\). Now, we need to determine if the series \(\sum_{k=1}^{\infty} \frac{1}{k^{2}}\) converges.

Determine if the comparison series converges

The series \(\sum_{k=1}^{\infty} \frac{1}{k^{2}}\) is a p-series with \(p = 2\). Since \(p > 1\), the p-series converges (by the p-series rule).

Conclusion

Since the comparison series \(\sum_{k=1}^{\infty} \frac{1}{k^{2}}\) converges and \(0 \leq \frac{\sin ^{2} k}{k^{2}} \leq \frac{1}{k^{2}}\) for all \(k\), the given series $$\sum_{k=1}^{\infty} \frac{\sin ^{2} k}{k^{2}}$$ also converges by the Comparison Test.

Key concepts

Comparison Test

The Comparison Test is a powerful tool used in determining the convergence of series. It is helpful when you can relate a given series to another series whose convergence is already known. This method involves comparing the terms of two series.

Here's how the test works:

• You need two series: \( \sum_{k=1}^{\infty} a_k\) and \(\sum_{k=1}^{\infty} b_k\).
• Ensure that for all terms, \(0 \leq a_k \leq b_k\).
• If \(\sum_{k=1}^{\infty} b_k\) converges, so does \(\sum_{k=1}^{\infty} a_k\).
• Conversely, if \(\sum_{k=1}^{\infty} a_k\) diverges, then so does \(\sum_{k=1}^{\infty} b_k\).

The Comparison Test is advantageous because it sometimes simplifies complex series evaluations. It provides a practical way to harness the behavior of well-studied series to infer the behavior of others.

P-Series

A p-series is a specific type of series which takes the form \(\sum_{k=1}^{\infty} \frac{1}{k^p}\), where \(p\) is a constant.

To determine convergence of a p-series, there is a simple rule to follow:

• If the constant \(p > 1\), the series converges.
• If \(0 < p \leq 1\), the series diverges.

For example, in the exercise, \(\sum_{k=1}^{\infty} \frac{1}{k^2}\) is a p-series where \(p = 2\). Since \(p > 1\), it converges. The simplicity of the p-series rule makes it a favorite for quick convergence checks in series.

Mathematical Series

A mathematical series is an expression representing the sum of the terms of a sequence. Understanding series is essential in calculus and further studies of mathematical analysis.

There are different types of series:

• Finite series: These have a limited number of terms.
• Infinite series: These extend indefinitely, often represented with the summation symbol \(\sum\).

Analyzing series helps in determining whether the sum is finite (convergent) or infinite (divergent). Different methods and tests, such as the Comparison Test, are utilized to evaluate the behavior of an infinite series.

Convergence Criteria

Convergence criteria are guidelines or rules that determine if a series will converge to a finite value. For infinite series, various tests help decide convergence. These criteria are part of essential mathematical tools that ensure comprehensively analyzing series behavior.

Common criteria and tests include:

• The Comparison Test: As discussed earlier, this test compares your series to another series with known convergence behavior.
• The Integral Test: Relates the series to an integral to check for convergence.
• The Ratio Test: Compares successive term ratios.
• The P-Series Test: Specific for series of the form \(\sum \frac{1}{k^p}\), useful for quick checks.

Choosing the right convergence test depends on the form and characteristics of the series in question. Each test has conditions under which it is most appropriately applied, helping mathematicians simplify and solve complex problems in calculus and analysis.