Problem 3 - The Calculus with Analytic Geometry

Question

Determine if the given sequence is increasing, decreasing, or not monotonic.\(\left\\{\frac{1}{n+\sin n^{2}}\right\\}\)

Step-by-Step Solution

Answer

The sequence is strictly decreasing.

Step 1 - Understand the Sequence

Examine the given sequence \(\frac{1}{n+\text{sin} n^{2}}\). This sequence is in the form of \(a_n = \frac{1}{n + \text{sin} n^2}\). Note that \( n \) represents the position in the sequence and \( \text{sin} n^2 \) represents the sine of \( n^2 \).

Step 2 - Analyze the Behavior of the Denominator

Since the sine function oscillates between -1 and 1, the expression \( n + \text{sin} n^2 \) ranges between \( n - 1 \) and \( n + 1 \). Thus, the denominator \( n + \text{sin} n^2 \) will grow as \( n \) increases.

Step 3 - Behavior of the Sequence as n Increases

Given that the denominator \( n + \text{sin} n^2 \) increases as \( n \) increases, the term \( a_n = \frac{1}{n + \text{sin} n^2} \) will decrease because the overall value is a fraction with a growing denominator.

Step 4 - Check for Monotonicity

A sequence is strictly decreasing if each term is less than the preceding term. To verify if \( a_n \) is strictly decreasing, observe that \( n + \text{sin} n^2 \) is always positive and increases monotonically for all positive integer values of \( n \). Hence, \( a_n \) decreases strictly.

Key Concepts

sequence behavior

Understanding sequence behavior is key to solving many mathematical problems. A sequence is simply a list of numbers in a specific order. The behavior of a sequence refers to how the terms of the sequence change as you move from one term to the next.
In this exercise, the given sequence is \(\frac{1}{n + \text{sin} n^2}\). To understand its behavior, we need to observe how each term changes as the variable \( n \) increases.
When analyzing a sequence, consider:

• The general form of the sequence
• The influence of each component on the sequence
• The overall trend as you move from one term to the next

In the sequence \(\frac{1}{n + \text{sin} n^2}\), as \( n \) grows, both the components \( n \) and \( \text{sin} n^2 \) affect the terms where \(\text{sin} n^2 \) oscillates between -1 and 1. As time progresses, the main trend is determined by the increasing value of \( n \).

monotonicity

Monotonicity refers to a sequence's behavior of either being entirely non-increasing or non-decreasing. In simple terms, a monotonic sequence will either always go up or always go down.
There are two types of monotonic sequences:

• Monotonically increasing (or non-decreasing): Each term is greater than or equal to the previous one.
• Monotonically decreasing (or non-increasing): Each term is less than or equal to the previous one.

To determine if a sequence is monotonic, we can compare each term with the prior one. For the sequence \(\frac{1}{n + \text{sin} n^2}\), we examined how the denominator \( n + \text{sin} n^2 \) changes. Since \( n + \text{sin} n^2 \) increases as \( n \) increases, the sequence value (which is \( \frac{1}{n + \text{sin} n^2} \)) will decrease because the fraction's denominator is increasing.
Therefore, this sequence is monotonically decreasing because \( \frac{1}{n + \text{sin} n^2} \) gets smaller as \( n \) becomes larger.

sine function

The sine function is a fundamental concept in trigonometry and mathematics. It describes a smooth periodic oscillation.
Key properties of the sine function include:

• It oscillates between -1 and 1.
• Its period is \( 2\boldsymbol{\text{π}} \), meaning it repeats every \( 2\boldsymbol{\text{π}} \) units.
• It is continuous and smooth.

In our given sequence \(\frac{1}{n + \text{sin} n^2}\), the sine function plays a critical role, although its impact is bounded.
Because \(\text{sin} n^2\) oscillates between -1 and 1, it causes the value of \( n + \text{sin} n^2 \) to lie between \( n - 1\) and \( n + 1\). However, as \( n \) grows larger, the effect of \(\text{sin} n^2 \) becomes less significant compared to the linear increase in \( n \). This results in the term \( n + \text{sin} n^2 \) growing steadily, leading to the overall decrease observed in the sequence \(\frac{1}{n + \text{sin} n^2}\).
Understanding how these functions behave helps in analyzing and predicting the sequence's overall trend.