Question

What does the divergence test tell us about the series \(\sum_{n=1}^{\infty} \cos \left(1 / n^{2}\right) ?\)

Short answer

The divergence test shows the series diverges, as the limit of the terms is 1.

Step-by-step solution

Define the Divergence Test

The Divergence Test states that for a series \( \sum_{n=1}^{\infty} a_n \), if the limit \( \lim_{n \to \infty} a_n eq 0 \), then the series diverges. If \( \lim_{n \to \infty} a_n = 0 \), the test is inconclusive.

Identify the sequence terms

For the series \( \sum_{n=1}^{\infty} \cos \left(1 / n^{2}\right) \), the sequence of terms is \( a_n = \cos \left(1 / n^{2}\right) \).

Find the limit of sequence terms

Calculate \( \lim_{n \to \infty} \cos \left(1 / n^{2}\right) \). As \( n \to \infty \), \( \frac{1}{n^2} \to 0 \). Therefore, \( \cos \left(1 / n^{2}\right) \to \cos(0) = 1 \).

Apply the Divergence Test

Since \( \lim_{n \to \infty} a_n = 1 eq 0 \), the Divergence Test tells us that the series \( \sum_{n=1}^{\infty} \cos \left(1 / n^{2}\right) \) diverges.

Key concepts

Infinite Series

An infinite series is a sum of an infinite sequence of terms. Imagine you start at a certain number and keep adding other numbers indefinitely. It is denoted as \( \sum_{n=1}^{\infty} a_n \), where \( a_n \) are the terms of the series.Let's consider a simple example: \( 1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \ldots \). This is an infinite series because we keep adding smaller and smaller fractions forever.
Whether or not such series add up to a finite number or drift off towards infinity, depend on their convergence. An important thing to keep in mind is that infinite series have a huge role to play in a vast variety of mathematical areas, from calculus to complex analysis, showcasing their depth and utility.

Convergence and Divergence

When dealing with infinite series, understanding convergence and divergence is vital. Convergence occurs when the sum of the infinite series approaches a specific limit:

• If the series \( \sum_{n=1}^{\infty} a_n \) converges, the terms get closer and closer to a certain value as \( n \) increases.
• For example, the series \( 1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \ldots \) actually converges to 2.

If the series \( \sum_{n=1}^{\infty} a_n \) diverges, its terms don’t get closer to any particular number.

An example of a diverging series is \( 1 + 1 + 1 + 1 + \ldots \), which keeps growing without bound.

In the context of the divergence test, a nonzero sequence limit indicates that a series diverges.

Sequence Limit

A sequence limit is the value that the terms \( a_n \) of a sequence get closer to as \( n \) gets very large. This is a fundamental concept used in analyzing series.When examining a sequence \( a_n \), if \( \lim_{n \to \infty} a_n \) exists, it can guide us in understanding the behavior of the related series.
Consider the example of \(\cos(1/n^2)\). As \( n \to \infty \), \(1/n^2\) tends toward zero, so \( \cos(1/n^2) \to \cos(0) = 1 \). Thus, the limit of our sequence terms is 1.
Keep in mind, for the convergence of a series, having sequence terms that limit to zero is necessary, but not sufficient. If the sequence terms do not limit to zero, the series definitely diverges.

Cosine Function

The cosine function, symbolized by \( \cos(x) \), is one of the basic trigonometric functions. It measures the horizontal coordinate of a point on the unit circle, relating to an angle \( x \) measured in radians.
Key properties include:

• The range of the cosine function is from -1 to 1.
• It is periodic, with a period of \( 2\pi \), meaning \( \cos(x) = \cos(x + 2\pi) \).
• At \( x = 0 \), \( \cos(0) = 1 \), indicating its maximum value.

When applied to small angles \( x \), the function approximately evaluates as 1, since \( \cos(x) \approx 1 \).
This property becomes apparent in our infinite series as \( n \) grows larger, making \( 1/n^2 \) effectively zero, and therefore \( \cos(1/n^2) \) hovers near 1. This insight showcases how the cosine function behaves with input values approaching zero.