Problem 65 - Calculus for Biology and Medicine

Question

\(\lim _{n \rightarrow \infty} a_{n}=a\). 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 is 0, and \( N = 100 \).

Step 1: Understand the Sequence

The sequence given is \( a_n = \frac{1}{\sqrt{n}} \). As \( n \) approaches infinity, \( a_n \) will approach 0, because the denominator grows larger, making the fraction smaller.

Step 2: Identify the Limit

From Step 1, we see that \( \lim_{n \to \infty} a_n = 0 \). Thus, the limit \( a \) is 0.

Step 3: Use the Epsilon Definition of a Limit

The definition requires finding \( N \) such that for \( n > N \), \( |a_n - a| < \epsilon \). Here \( a = 0 \), so the requirement becomes \( |\frac{1}{\sqrt{n}} - 0| < 0.1 \). Simplifying, this is \( \frac{1}{\sqrt{n}} < 0.1 \).

Step 4: Solve the Inequality

Rearrange the inequality \( \frac{1}{\sqrt{n}} < 0.1 \). This implies \( \sqrt{n} > 10 \). Squaring both sides gives \( n > 100 \). Therefore, for all \( n > 100 \), \( |\frac{1}{\sqrt{n}} - 0| < 0.1 \) is satisfied.

Step 5: Conclusion for \( N \)

The value of \( N \) that satisfies the condition is \( 100 \). For all \( n > 100 \), \( |a_n - a| < 0.1 \) holds true.

Key Concepts

sequence limits

Understanding the concept of sequence limits is crucial in calculus and analysis. A sequence is an ordered list of numbers that usually follow a specific formula or pattern. In mathematical terms, a sequence is said to have a limit if the numbers in the sequence approach a certain value as you progress further into the sequence. Think of it like a car rolling down a hill and coming to a gradual stop at a specific point. The point where the car stops is akin to the limit of the sequence. For the sequence given in the exercise, which is \[ a_n = \frac{1}{\sqrt{n}} \] as \(n\) becomes larger and larger, the terms of the sequence \(a_n\) get closer and closer to 0. Since the formula for \(a_n\) includes \(1/\sqrt{n}\), as \(n\) increases, \(\sqrt{n}\) grows significantly, making \(1/\sqrt{n}\) smaller and approaching zero. Thus, the limit \(a\) of this sequence as \(n\rightarrow\infty\) is indeed 0, which means \[ \lim_{n\rightarrow\infty} a_n = 0. \] Understanding this limit is critical as it simplifies how we anticipate the behavior of the sequence as it progresses.

inequality solving

Inequality solving often plays a key role in determining the behavior of sequences and functions within limits. When we talk about an epsilon-delta definition or indeed any limit, simplifying inequalities allows us to pinpoint when a sequence reaches a state that satisfies a condition for convergence. In the context of the epsilon-delta definition:

• \(\epsilon\) represents the small positive number that signifies how close the sequence must get to the limit \(a\).
• We must solve inequalities to find \(N\), after which all terms \(a_n\) are within \(\epsilon\) of \(a\).

For our exercise, we started with the inequality \[ \left|a_n - a\right| = \left|\frac{1}{\sqrt{n}} - 0\right| < 0.1. \] We need to solve \(\frac{1}{\sqrt{n}} < 0.1\) to find \(N\), which involves manipulating the inequality:

• Take the reciprocal to get \(\sqrt{n} > 10\).
• Square both sides giving \(n > 100\).

Thus, it is for all \(n > 100\) that \(a_n\) is close enough to the limit such that the distance from \(a\) is less than \(\epsilon\). This shows both spatial understanding and algebraic manipulation to reach a solution.

asymptotic behavior

Grasping asymptotic behavior helps us understand how functions and sequences behave as they approach infinity, or a boundary condition. When discussing a sequence like \(a_n = \frac{1}{\sqrt{n}}\), the asymptotic behavior describes how \(a_n\) behaves as \(n\) becomes very large. Think of a sequence getting closer to horizontal as \(n\) increases.

• The graph of this particular sequence will show the terms decreasing sharply at first, but the rate of decrease slows down as \(n\) increases further.
• This decreasing behavior reflects the sequence asymptotically approaching zero.

In mathematical analysis, we say \(a_n\) is asymptotically equivalent to zero because as \(n\rightarrow\infty\), with every increase in \(n\), the changes to \(a_n\) become less significant.This concept is vital when predicting long-term trends in calculus, and it helps identify more about the behavior of functions especially when the precise limit is zero or when the function tends towards a specific value as one of the parameters increases infinitely. Understanding asymptotic behavior equips us to anticipate the result of functions and sequences beyond their immediate calculations.