Problem 57 - Calculus for Biology and Medicine

Question

Formal Definition of Limits: \lim _{n \rightarrow \infty} a_{n}=a .\( Find the limit \)a\(, and determine \)N\( so that \)\left|a_{n}-a\right|<\epsilon\( for all \)n>N\( for the given value of \)\epsilon$. $$ a_{n}=\frac{1}{\sqrt{n}}, \epsilon=0.1 $$

Step-by-Step Solution

Answer

The limit \( a \) is 0, and \( N = 100 \) ensures \( |a_n - a| < \epsilon \).

Step 1: Identify the Limit

To find the limit of the sequence \( a_n = \frac{1}{\sqrt{n}} \) as \( n \to \infty \), observe that \( \sqrt{n} \to \infty \) as \( n \to \infty \), so \( \frac{1}{\sqrt{n}} \to 0 \). Therefore, the limit \( a \) is 0.

Step 2: Set Up the Inequality

We want to satisfy the condition \( |a_n - a| < \epsilon \). Since \( a = 0 \), the inequality becomes \( | \frac{1}{\sqrt{n}} - 0 | < 0.1 \), which simplifies to \( \frac{1}{\sqrt{n}} < 0.1 \).

Step 3: Solve for n

Solve the inequality \( \frac{1}{\sqrt{n}} < 0.1 \). Rearrange to get \( \sqrt{n} > 10 \). By squaring both sides, we have \( n > 100 \).

Step 4: Determine N

From the inequality \( n > 100 \), we can choose \( N = 100 \). For all \( n > N \), the original inequality \( |a_n - a| < 0.1 \) is satisfied.

Key Concepts

Formal Definition of Limits

In the fascinating world of calculus, the concept of limits is foundational. It's essential for understanding how functions behave as inputs approach a particular point, often infinity. When dealing with sequences, the formal definition of a limit, denoted by \( \lim_{n \rightarrow \infty} a_{n} = a \), asserts that as \( n \) (a variable typically representing natural numbers) increases without bound, the value of \( a_n \) gets arbitrarily close to \( a \).
This means that for any arbitrarily small positive number \( \epsilon \), there exists some positive integer \( N \) such that for all \( n > N \), the distance between \( a_n \) and \( a \), expressed as \( |a_n - a| \), is less than \( \epsilon \).
In practical terms, this defines the process by which a sequence approaches a fixed value, illustrating convergence in a clear, mathematically rigorous manner.

Sequence Convergence

Understanding convergence is crucial when working with sequences in calculus. A sequence \( \{a_n\} \) is said to converge to a limit \( a \) if the terms of the sequence become arbitrarily close to \( a \) as \( n \) becomes very large. For example, in the sequence \( a_n = \frac{1}{\sqrt{n}} \), as \( n \) increases, \( \sqrt{n} \) becomes very large, causing \( \frac{1}{\sqrt{n}} \) to get very close to 0.

• Convergence indicates stability, as the sequence settles towards a fixed value.
• This behavior is crucial in mathematical analysis, allowing predictions and calculations concerning infinite processes to be finite and understandable.

Determining the limit, as we explored, involves observing the sequence's trend as the input grows indefinitely. Convergence, in essence, translates complex infinite behavior into something manageable and bound by logic.

Inequality Solving

Inequality solving is a fundamental mathematical skill, allowing us to find limits effectively. In the context of determining limits for sequences, it's about ensuring that the sequence terms fall within a specified range. When tasked with finding \( N \) such that \( |a_n - a| < \epsilon \) for all \( n > N \), solving an inequality provides the answer.
Taking the sequence \( a_n = \frac{1}{\sqrt{n}} \) as an example, once we determined that it converges to 0, we set up the inequality \( \frac{1}{\sqrt{n}} < 0.1 \). This moved us forward by solving \( \sqrt{n} > 10 \), revealing \( n > 100 \). Therefore, choosing any \( N \geq 100 \) suffices.

• Inequality solving bridges logic and numerical value, providing a systematic approach to finding critical values like \( N \).
• Mastering this technique is essential for tackling limit problems across calculus.

By systematically breaking down the inequality, you understand not just what \( N \) must be but why, reinforcing broader mathematical understanding.