Problem 58 - 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$. $$ a_{n}=\frac{1}{\sqrt{n}}, \epsilon=0.05 $$

Step-by-Step Solution

Answer

The limit is 0, and \(N = 400\) for \(\epsilon = 0.05\).

Step 1: Identify the Expected Limit

We need to find the limit \(a\) as \(n\) approaches infinity for the sequence \(a_n = \frac{1}{\sqrt{n}}\). As \(n\) becomes very large, \(\frac{1}{\sqrt{n}}\) approaches 0. Thus, the limit \(a\) is 0.

Step 2: Implement the Epsilon-Delta Definition

For the given \(\epsilon = 0.05\), we need to find \(N\) such that \(\left| \frac{1}{\sqrt{n}} - 0 \right| < 0.05\) for all \(n > N\). This simplifies to \(\frac{1}{\sqrt{n}} < 0.05\).

Step 3: Solve Inequality for N

To solve \(\frac{1}{\sqrt{n}} < 0.05\), start by rearranging: \(\sqrt{n} > \frac{1}{0.05} = 20\). Squaring both sides gives \(n > 400\). Therefore, \(N = 400\) satisfies the condition.

Key Concepts

Epsilon-Delta Definition

The epsilon-delta definition is fundamental in understanding how limits work in calculus. This precise definition allows us to say, "The limit of a sequence is \(a\) as \(n\) approaches infinity," with mathematical rigor.When we speak about limits using the epsilon-delta definition, we mean that for every very small positive number \(\epsilon\) (like 0.05), there is a corresponding larger number \(N\) such that for all terms in the sequence after \(N\), the terms are within \(\epsilon\) of the limit \(a\). In simpler terms:- Choose a small \(\epsilon\), which dictates how close we want the sequence terms \(a_n\) to be to the limit \(a\).- Find \(N\) so that if \(n > N\), then \(a_n\) is practically hugging the limit within our demanded \(\epsilon\). This idea of making terms as closely packed around the limit is crucial in calculus and helps in understanding the precision of sequences approaching a particular point.

Infinity Limit

In calculus, limits that involve infinity are essential for understanding how functions behave as values grow larger and larger. An infinity limit shows us what the result would be as an input variable tends toward infinitely large numbers.For the sequence \(a_n = \frac{1}{\sqrt{n}}\), as \(n\) tends toward infinity, the expression \(\frac{1}{\sqrt{n}}\) grows closer to 0. This is because the square root of \(n\) increases, causing the entire fraction to decrease towards zero. Thus, the limit \(a\) is 0 in this case.Understanding limits at infinity:- Shows long-term behavior of sequences or functions.- Helps clarify whether terms decrease to zero or perhaps increase without bound.Studying limits at infinity provides a broader view of how sequences play out over vast ranges, which is important for applications in science and engineering.

Sequence Convergence

Sequence convergence is a concept that defines whether a sequence approaches a certain value as the index progresses to infinity. If a sequence converges, it means its terms become closer and closer to a specific limit.In the problem, the sequence was given by \(a_n = \frac{1}{\sqrt{n}}\). As \(n\) increases, the sequence's terms get nearer to 0, showing its convergence to the limit 0. Here's how we approach this:- Expression \(a_n = \frac{1}{\sqrt{n}}\) reveals that as \(n\) becomes large, \(a_n\) heads towards 0.- Convergence uses the epsilon-delta definition to assure that after a certain \(N\), all terms are within a chosen distance, \(\epsilon\), from the limit.Understanding whether a sequence converges helps in determining the behavior and prediction of the sequence in real-world scenarios where such mathematical abstractions are useful.