Problem 69 - 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 \(n>N\). $$ \lim _{n \rightarrow \infty} \frac{n+1}{n}=1 $$

Step-by-Step Solution

Answer

For any \(\epsilon > 0\), choosing \(N = \lceil \frac{1}{\epsilon} \rceil\) ensures \(\left| \frac{n+1}{n} - 1 \right| < \epsilon\) for all \(n > N\).

Step 1: Understand the Limit Definition

Given the sequence \(a_n = \frac{n+1}{n}\), we want to show \(\lim_{n \to \infty} a_n = 1\) using the definition of limits. The definition states that for every \(\epsilon > 0\), there exists an \(N\) such that \(|a_n - a| < \epsilon\) for all \(n > N\). Here, \(a = 1\).

Step 2: Express the Absolute Difference

Calculate \(|a_n - a|\) where \(a_n = \frac{n+1}{n}\) and \(a = 1\). This gives: \[|a_n - a| = \left|\frac{n+1}{n} - 1\right|\] Simplify inside the absolute value: \[= \left|\frac{n+1-n}{n}\right| = \left|\frac{1}{n}\right| = \frac{1}{n}\] Since \(n\) is a positive integer, we have \(|a_n - a| = \frac{1}{n}\).

Step 3: Find Suitable N for Given Epsilon

We want to ensure that \(\frac{1}{n} < \epsilon\). To find \(N\) such that this holds for all \(n > N\), solve the inequality \(\frac{1}{n} < \epsilon\) by rearranging to find \(n > \frac{1}{\epsilon}\). Thus, choose \(N = \lceil \frac{1}{\epsilon} \rceil\).

Step 4: Verify Condition from Definition

For \(n > N = \lceil \frac{1}{\epsilon} \rceil\), by setting \(n > \frac{1}{\epsilon}\), it follows that \(\frac{1}{n} < \epsilon\). Thus, for \(n > N\), \(|a_n - a| = \frac{1}{n} < \epsilon\), satisfying the limit definition.

Key Concepts

Sequence Convergence

Sequence convergence is a fundamental concept in calculus and analysis. It describes how the terms of a sequence approach a specific value, known as the limit, as the index (usually denoted as \(n\)) becomes infinitely large. When we say a sequence \(a_n\) converges to a value \(a\), it implies that the difference between \(a_n\) and \(a\) becomes arbitrarily small as \(n\) increases. Key aspects of sequence convergence include:

• Approaching a Limit: The sequence elements \(a_n\) get increasingly close to the limit \(a\) as the sequence progresses.
• Uniqueness: A sequence can have at most one limit, ensuring consistency in its behavior as it converges.
• Criterion: For convergence, for every positive number \(\epsilon\), there should be an \(N\) such that whenever \(n > N\), the absolute value of the difference \(|a_n - a| < \epsilon\).

Understanding sequence convergence is crucial for dealing with infinite processes and ensuring continuity in mathematical analysis.

Epsilon-Delta Definition

The epsilon-delta definition is a precise, mathematical way to define limits, which is pivotal in understanding calculus. It involves two critical components: \(\epsilon\) and \(\delta\). In the context of sequence limits, this definition works as follows:

• For every small positive number \(\epsilon\) (representing how close you want your sequence terms to be to the limit), there is a corresponding index \(N\). This \(N\) ensures that all sequence terms beyond this point are within an \(\epsilon\) distance from the limit.
• Symbolically, this is captured by: for every \(\epsilon > 0\), there exists an \(N\) such that if \(n > N\), then \(|a_n - a| < \epsilon\).

By applying the epsilon-delta definition, we concretely verify the behavior of sequences as they approach their limit. This is essential in proving convergence rather than just estimating or guessing. It's like setting a tangible rule for how close matters should be to say they meet the limit criteria.

Absolute Value Inequality

Absolute value inequalities are integral in expressing how far a term is from a specific value, which is crucial when dealing with convergence. The absolute value function, \(|x|\), denotes the non-negative distance of a number \(x\) from zero on the number line.When using absolute value inequalities in limits, we often face expressions like \(|a_n - a| < \epsilon\), which indicate that the difference between \(a_n\) and \(a\) is less than the chosen positive value \(\epsilon\). This helps in validating that the terms of the sequence are close enough to the limit for large \(n\). Some aspects to remember include:

• Understanding \(|x|\): The expression represents distance, which is always non-negative.
• Inequalities such as \(|a_n - a| < \epsilon\): These are used to set precise limits and thresholds in calculus and analysis, ensuring terms of sequences or functions are practically close to a target value.

By mastering absolute value inequalities, you gain a robust tool in handling and calculating how sequences and functions behave as they approach specific values.