Problem 68 - Calculus for Biology and Medicine

Question

Formal Definition of Limits: Use the formal definition of limits to show that \(\lim _{n \rightarrow \infty} a_{n}=a ;\) that is, find \(N\) such that for every \(\epsilon>0\), there exists an \(N\) such that \(\left|a_{n}-a\right|<\epsilon\) whenever \(n>N\). $$ \lim _{n \rightarrow \infty} \frac{1}{n^{2}+1}=0 $$

Step-by-Step Solution

Answer

The limit is proven using \( N = \lceil \sqrt{\frac{1}{\epsilon} - 1} \rceil \), showing that \( \lim_{n \to \infty} \frac{1}{n^2 + 1} = 0 \).

Step 1: Identifying the Limit

We are asked to prove that \( \lim_{n \to \infty} \frac{1}{n^2 + 1} = 0 \). Here, the sequence \( a_n = \frac{1}{n^2 + 1} \) should converge to 0 as \( n \to \infty \). This means for every \( \epsilon > 0 \), there should be an \( N \) such that \( |a_n - 0| < \epsilon \) whenever \( n > N \).

Step 2: Expressing the Inequality

The expression \( |a_n - 0| \) simplifies to \( a_n = \frac{1}{n^2 + 1} \). We need \( \frac{1}{n^2 + 1} < \epsilon \). Our goal is to find \( N \) such that this holds true whenever \( n > N \).

Step 3: Solving the Inequality

The inequality \( \frac{1}{n^2 + 1} < \epsilon \) can be solved by manipulating it to find conditions on \( n \). It can be rewritten as \[ n^2 + 1 > \frac{1}{\epsilon} \]. Hence, \[ n^2 > \frac{1}{\epsilon} - 1 \].

Step 4: Finding N

To find \( N \), solve the inequality \( n^2 > \frac{1}{\epsilon} - 1 \). We take the square root to get \( n > \sqrt{\frac{1}{\epsilon} - 1} \). Choose \( N = \lceil \sqrt{\frac{1}{\epsilon} - 1} \rceil \) such that whenever \( n > N \), \( \frac{1}{n^2 + 1} < \epsilon \).

Step 5: Conclusion

For every \( \epsilon > 0 \), if \( N = \lceil \sqrt{\frac{1}{\epsilon} - 1} \rceil \), then \( n > N \) implies \( \frac{1}{n^2 + 1} < \epsilon \). Thus, by the formal definition of limits, \( \lim_{n \to \infty} \frac{1}{n^2 + 1} = 0 \).

Key Concepts

Convergence

Convergence is a fundamental idea in calculus, especially when dealing with sequences and series. A sequence is said to converge if its terms approach a specific value as the number of terms grows infinitely. In simpler terms, when you keep adding terms to the sequence and it gets closer and closer to a particular number, it is converging to that value.
For example, in the exercise provided, we analyzed the sequence \(a_n = \frac{1}{n^2 + 1}\). As \(n\) gets larger and larger, the value of this sequence gets closer to 0. This means that \(a_n\) converges to 0.
Some key points about convergence include:

• The limit of a convergent sequence is the value it approaches as \(n\) becomes infinitely large.
• A sequence may not converge if it oscillates or increases without bound.
• When discussing convergence, we often talk about the "limit" to define what the sequence approaches.

Understanding convergence can help you comprehend how mathematical functions behave over large or tiny intervals. It's like predicting the behavior of a sequence as you increase or decrease its input values.

Epsilon-Delta Definition

The Epsilon-Delta definition of limits is a rigorous way to demonstrate convergence in calculus. This formal definition ensures precision in showing that a sequence or function converges to a specific limit. Let's break it down to comprehend it better.
The definition states that for any small number \(\epsilon > 0\), there must exist a number \(N\) such that for all \(n > N\), the difference between the sequence \(a_n\) and the limit \(a\) is less than \(\epsilon\). In mathematical terms, this is written as:\[|a_n - a| < \epsilon\]when \(n > N\).
This implies that you can make the sequence terms as close as you desire to the limit, by choosing a sufficiently large \(N\).
In practice, using this definition helps us prove that \(\lim_{n \to \infty} \frac{1}{n^2 + 1} = 0\). For any chosen \(\epsilon\), we can find a corresponding \(N\) such that for all \(n > N\), \(\frac{1}{n^2 + 1} < \epsilon\). By taking the necessary steps, we showed how this works for the given problem.

Infinite Limits

Infinite limits are a fascinating topic involving sequences or functions that grow without bounds. Instead of approaching a fixed number, these entities tend towards positive or negative infinity.
In the context of sequences as they increase, we discuss the idea of the limit approaching zero. For the sequence \(\frac{1}{n^2 + 1}\), as \(n\) approaches infinity, the terms get smaller and closer to zero, demonstrating that the limit is zero. However, infinite limits can be existing in other terms such as when we consider \(\lim_{n \to \infty} n = \infty\).
Consider the points to understand infinite limits better:

• When a sequence grows without bound, we describe it using the term 'infinite limit.'
• Infinite limits aren't about reaching a specific number but rather 'flying off' beyond any given value.
• Understanding infinite limits is crucial for grasping the behavior of functions over very large inputs or very small ranges approaching an endpoint.

In sum, infinite limits provide insight into the vast reaches of mathematical sequences and functions, exploring their unfathomable bounds.