Problem 73 - Calculus for Biology and Medicine

Question

In Problems \(73-78\), 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 \(n>N\). $$ \lim _{n \rightarrow \infty} \frac{3}{n}=0 $$

Step-by-Step Solution

Answer

For every ε>0, choose N=⌈3/ε⌉; then |3/n - 0| < ε when n>N.

Step 1: Express the Limit Definition

The limit \(\lim_{n \rightarrow \infty} \frac{3}{n} = 0\) \ means that for every \ \epsilon > 0\, there exists a positive integer \(N\) such that \(\left|\frac{3}{n} - 0\right| < \epsilon\) whenever \(n > N\). This translates to \(\frac{3}{n} < \epsilon\) for \(n > N\).

Step 2: Solve for n in Terms of ε

From the inequality \(\frac{3}{n} < \epsilon\), we multiply both sides by \(n\) and divide by \(\epsilon\) to obtain \(3 < n\epsilon\). This gives us \(n > \frac{3}{\epsilon}\).

Step 3: Choose N in Terms of ε

From the previous step, we found that for \(n > \frac{3}{\epsilon}\), the inequality holds. Choose \(N\) to be equal to \(\lceil \frac{3}{\epsilon} \rceil\), where \(\lceil x \rceil\) represents the ceiling function to ensure \(N\) is an integer.

Step 4: Conclude the Limit Proof

For every \(\epsilon > 0\), we have found \(N = \lceil \frac{3}{\epsilon} \rceil\) such that for all \(n > N\), \(\left|\frac{3}{n} - 0\right| < \epsilon\). This confirms that the limit \(\lim_{n \rightarrow \infty} \frac{3}{n} = 0\) by the definition of a limit.

Key Concepts

Formal Definition of Limits

The formal definition of limits helps us precisely express the idea of a sequence approaching a specific value. When we say a sequence \(a_n\) approaches \(a\) as \(n\) approaches infinity, we mean that the values of \(a_n\) can get arbitrarily close to \(a\) as \(n\) becomes very large. So, for every positive number \(\epsilon\), no matter how small, there exists an integer \(N\) such that all terms of the sequence beyond the \(N^{th}\) term lie within \(\epsilon\) of \(a\).
This means:

• For any error margin \(\epsilon > 0\), we can find an integer \(N\).
• All sequence terms \(a_n\) with \(n > N\) will satisfy \(|a_n - a| < \epsilon\).

This definition ensures that \(a\) is the limit of \(a_n\). By formalizing limits in this way, we gain a tool to tackle problems involving convergence and behavior of sequences.

Epsilon-Delta Definition

The epsilon-delta definition is an extension of the formal limits concept, primarily used to define limits of functions. It adds rigor by specifying conditions under which a particular function value can be "approached" by outputs related to nearby inputs.
For functions \(f(x)\) approaching \(L\) as \(x\) approaches \(c\), this definition is stated as:

• For every \(\epsilon > 0\), there is a \(\delta > 0\) such that whenever \(0 < |x - c| < \delta\), it follows that \(|f(x) - L| < \epsilon\).

This means that for any tiny "tolerance" value \(\epsilon\), we can squeeze \(f(x)\) values close to \(L\) by adjusting inputs to be within \(\delta\) range from \(c\). The epsilon-delta approach provides a detailed groundwork for understanding "closeness" in mathematical functions.

Infinite Sequences

Infinite sequences are ordered lists of numbers, typically with no end. They are represented as \(a_1, a_2, a_3, \ldots\). A key interest is identifying the sequence's behavior as the number of terms extends indefinitely.
Some key points include:

• If a sequence converges to a limit \(L\), for every \(\epsilon > 0\), there exists an \(N\) such that all terms beyond the \(N^{th}\) are within \(\epsilon\) of \(L\).
• Not all sequences converge; some are divergent and won't settle around any single value.

Working with infinite sequences involves assessing their limits, growth rate, or periodicity. Grasping these aspects is essential to tackle broader mathematical analyses and applications.