Question

Use the formal definition of the limit of a sequence to prove the following limits. $$\lim _{n \rightarrow \infty} \frac{c n}{b n+1}=\frac{c}{b}, \text { for real numbers } c > 0 \text { and } b > 0$$

Short answer

Question: Prove that, for real numbers c > 0 and b > 0, the limit of the sequence $$\frac{cn}{bn+1}$$ as n approaches infinity is $$\frac{c}{b}$$ using the formal definition of the limit of a sequence. Answer: To prove this, we showed that for any given positive real number ε, there exists a positive integer N such that for all natural numbers n > N, the absolute difference between the sequence's term and the limit is less than ε. We simplified the expression inside the absolute value and isolated n on one side of the inequality. Then, we found an appropriate N for any given ε and showed that the inequality holds for all n > N. Therefore, by the formal definition of the limit of a sequence, the limit of $$\frac{cn}{bn+1}$$ as n approaches infinity is $$\frac{c}{b}$$.

Step-by-step solution

State the formal definition of the limit of a sequence

We want to prove that for any positive real number ε, there exists a positive integer N such that for all natural numbers n > N, $$\left|\frac{cn}{bn+1}-\frac{c}{b}\right| < \epsilon$$

Manipulate the expression inside the absolute value

To simplify the expression inside the absolute value, we find a common denominator: $$\left|\frac{cn}{bn+1}-\frac{c}{b}\right| = \left|\frac{cbn-c(bn+1)}{b(bn+1)}\right|$$ Now, simplify the numerator: $$\left|\frac{cbn-c(bn+1)}{b(bn+1)}\right| = \left|\frac{-c}{b(bn+1)}\right|$$ Since c and b are positive, we can ignore the absolute value and just focus on the following inequality: $$\frac{c}{b(bn+1)} < \epsilon$$

Isolate n on one side of the inequality

We want to find a condition on n which guarantees that the inequality holds. To do this, isolate n on one side of the inequality. First, multiply both sides by \(b(bn+1)\): $$c < \epsilon b(bn+1)$$ Now, divide both sides by c and b: $$\frac{1}{\epsilon} > bn + 1$$ Then, subtract 1 from both sides: $$\frac{1}{\epsilon} - 1 > bn$$ Finally, divide both sides by b: $$\frac{\frac{1}{\epsilon} - 1}{b} > n$$

Find an appropriate N and show that the inequality holds

Note that we can now find an appropriate N for any given ε. To show that the inequality holds for all n > N, we can choose $$N = \left\lceil\frac{\frac{1}{\epsilon} - 1}{b}\right\rceil$$ Where \(\lceil x \rceil\) denotes the smallest integer greater than or equal to x. Then, for all n > N, we must have: $$n > \frac{\frac{1}{\epsilon} - 1}{b}$$ and therefore, $$\frac{c}{b(bn+1)} < \epsilon$$ Thus, by the formal definition of the limit of a sequence, we have shown that: $$\lim _{n \rightarrow \infty} \frac{cn}{bn+1}=\frac{c}{b}, \text { for real numbers } c > 0 \text { and } b > 0$$

Key concepts

Epsilon-Delta Definition

The epsilon-delta definition is a fundamental approach used to rigorously define the concept of limits, crucial for calculus and analysis principles. When we talk about limits in the context of sequences, we're often referring to how the terms of the sequence behave as they extend indefinitely. The goal is to create precise conditions under which a sequence approaches a specific limit as the terms increase.

In this case, the limit of a sequence \(\lim_{n \to \infty} a_n = L\) implies that for any arbitrarily chosen positive number \(\epsilon\) (no matter how small), there exists a particular natural number, known as \(N\). Beyond this threshold, for every natural number \(n > N\), the difference between the sequence's terms \(a_n\) and the limit \(L\) is less than \(\epsilon\). Put more simply, as \(n\) becomes very large, the terms of the sequence get closer and closer to \(L\).

• \(\epsilon\) represents how close we want the sequence terms to the limit.
• \(N\) is the point beyond which all terms of the sequence stay within this close range.

This definition underpins our ability to conclude definitively whether a sequence converges to a particular number.

Infinite Sequences

Infinite sequences are an ordered list of numbers designed to go on endlessly. In mathematics, they're an essential concept because they represent a process or pattern evolving over an infinite domain. These sequences come in different types and classes, but when it comes to analyzing limits, we're interested in how these infinite sequences behave as they tend toward infinity.

Sequences can diverge, approach a finite limit, or even oscillate between values. Convergence and divergence describe this behavior:

• Convergent Sequence: The sequence approaches a specific, finite number as \(n\) becomes infinitely large.
• Divergent Sequence: The sequence does not settle toward any single value, possibly growing without bound or lacking a predictable pattern.

Our example, \(\frac{cn}{bn+1}\), is convergent, tending towards \(\frac{c}{b}\) as \(n\) increases. Practically, this means evaluating the limit offers insight into the pattern's long-term behavior. Understanding these large \(n\) behaviors helps in fields requiring advanced prediction and analysis such as physics and economics.

Limit Theorem Proof

The proof of a limit theorem, such as showing \(\lim_{n \to \infty} \frac{c n}{b n+1}=\frac{c}{b}\), involves demonstrating, under general conditions, that a sequence approaches a particular limit. For this, we use the epsilon-delta definition.

To conduct this proof:1. **Begin with the definition:** Express the absolute difference between the sequence and its alleged limit. In this case, \( \left| \frac{cn}{bn+1} - \frac{c}{b} \right| \).2. **Simplify the expression:** Use basic algebra to manipulate the formula to a state where you can isolate \(n\) beyond the mathematical inequalities.3. **Establish bounds on \(n\):** Typically, you'll need to show that beyond some threshold \(N\), all sequence terms fit within the chosen \(\epsilon\) range.4. **Select \(N\):** Ensure \(N\) is chosen such that for all \(n > N\), the terms comply with the requirements. In our example, setting \(N = \left\lceil \frac{\frac{1}{\epsilon} - 1}{b} \right\rceil\) achieves this.This careful process confirms the sequence reaches the desired limit as \(n\) approaches infinity, utterly relying on the precision of the epsilon-delta method. Such proofs ensure the assumptions about infinite sequences abide strictly by logical allowances, enabling their reliable use in real-world calculations and models.