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Chapter 9: Problem 76

Which of the sequences \(\left\\{a_{n}\right\\}\) converge, and which diverge? Find the limit of each convergent sequence. $$a_{n}=\sinh (\ln n)$$

Short Answer

Expert verified
The sequence diverges as it approaches infinity.

Step by step solution

01

Understand the Sequence

The given sequence is \( a_n = \sinh(\ln n) \). To determine convergence, we need to analyze the behavior of \( \sinh(\ln n) \) as \( n \to \infty \). Recall that the hyperbolic sine function \( \sinh(x) = \frac{e^x - e^{-x}}{2} \).
02

Substitute Inside Hyperbolic Sine

The next step involves substituting \( x = \ln n \) into the hyperbolic sine function. Thus, \( \sinh(\ln n) = \frac{e^{\ln n} - e^{-\ln n}}{2} \).
03

Simplify the Expression

Now simplify the expression: \( e^{\ln n} = n \) and \( e^{-\ln n} = \frac{1}{n} \), since \( e^{-x} = \frac{1}{e^x} \). Thus, \( \sinh(\ln n) = \frac{n - \frac{1}{n}}{2} = \frac{n^2 - 1}{2n} \).
04

Evaluate the Limit

To find the limit, compute \( \lim_{n \to \infty} \frac{n^2 - 1}{2n} \). This simplifies to \( \lim_{n \to \infty} \left( \frac{n^2}{2n} - \frac{1}{2n} \right) = \lim_{n \to \infty} \left( \frac{n}{2} - \frac{1}{2n} \right) \).
05

Solve the Limit

As \( n \to \infty \), \( \frac{n}{2} \to \infty \) and \( \frac{1}{2n} \to 0 \). Thus, \( \lim_{n \to \infty} \left( \frac{n}{2} - \frac{1}{2n} \right) = \infty \).
06

Conclusion on Convergence

Since the limit of the sequence as \( n \to \infty \) is \( \infty \), the sequence \( a_n = \sinh(\ln n) \) diverges.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Hyperbolic Functions
Hyperbolic functions, much like trigonometric functions, are important in mathematics and define the relationship between exponential functions. One common hyperbolic function is the hyperbolic sine, noted as \( \sinh(x) \). The formula for \( \sinh(x) \) is given by:\[\sinh(x) = \frac{e^x - e^{-x}}{2}\]This equation captures the difference between the exponential function \( e^x \) and its inverse \( e^{-x} \), divided by two. Studying these functions helps us understand growth patterns and solve differential equations that appear in physics and engineering.
We used the hyperbolic sine function in the problem to explore sequence behavior involving natural logarithms, transforming it using basic properties of logarithms and exponentials.
Limits
In mathematics, a limit helps us analyze certain behaviors in sequences and functions. It tells us what value a function (or sequence) approaches as the input grows or shrinks. Understanding limits is a fundamental aspect of calculus. Particularly, when chasing the limit of a sequence \( a_n \) as \( n \) tends towards infinity, it tells us if the sequence approaches a specific number or not.
For the sequence \( a_n = \sinh(\ln n) \), we evaluated the limit to determine its behavior. After simplifying, the sequence reduced to \( \frac{n^2 - 1}{2n} \), and further as \( \frac{n}{2} - \frac{1}{2n} \). As \( n \to \infty \), the term \( \frac{n}{2} \to \infty \) while \( \frac{1}{2n} \to 0 \). Therefore, the sequence does not approach a specific number, leading to the conclusion about its divergence.
Divergent Sequence
A sequence diverges when it fails to settle down to any particular limit as it progresses to infinity. Simply put, a divergent sequence keeps increasing or decreasing with no bound or oscillates without converging to a single value. Divergence indicates lack of stability within a sequence over time.
  • If a sequence, such as \( a_n = \sinh(\ln n) \), veers off to infinity without approaching any number, it means it diverges.
  • The steps we followed demonstrated this by simplifying the sequence to \( \frac{n}{2} - \frac{1}{2n} \) and noticing that the primary component, \( \frac{n}{2} \), led it to grow unbounded.
Understanding divergence is crucial for identifying non-settling processes in mathematical series and functions, which are often analyzed in areas involving infinite series, physics, and engineering problems.

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Most popular questions from this chapter

What happens if you add a finite number of terms to a divergent series or delete a finite number of terms from a divergent series? Give reasons for your answer.

Which of the sequences converge, and which diverge? Give reasons for your answers. $$a_{n}=\left((-1)^{n}+1\right)\left(\frac{n+1}{n}\right)$$

Neither the Ratio Test nor the Root Test helps with \(p\) -series. Try them on $$ \sum_{n=1}^{\infty} \frac{1}{n^{p}} $$ and show that both tests fail to provide information about convergence.

Use a CAS to perform the following steps for the sequences. a. Calculate and then plot the first 25 terms of the sequence. Does the sequence appear to be bounded from above or below? Does it appear to converge or diverge? If it does converge, what is the limit \(L ?\) b. If the sequence converges, find an integer \(N\) such that \(\left|a_{n}-L\right| \leq 0.01\) for \(n \geq N .\) How far in the sequence do you have to get for the terms to lie within 0.0001 of \(L ?\) $$a_{n}=\sqrt[n]{n}$$

Pythagorean triples \(\quad\) A triple of positive integers \(a, b,\) and \(c\) is called a Pythagorean triple if \(a^{2}+b^{2}=c^{2} .\) Let \(a\) be an odd positive integer and let $$b=\left\lfloor\frac{a^{2}}{2}\right\rfloor \quad \text { and } \quad c=\left\lceil\frac{a^{2}}{2}\right\rceil$$ be, respectively, the integer floor and ceiling for \(a^{2} / 2\). a. Show that \(a^{2}+b^{2}=c^{2} .\) (Hint: Let \(a=2 n+1\) and express \(b \text { and } c \text { in terms of } n .)\) b. By direct calculation, or by appealing to the accompanying figure, find $$\lim _{a \rightarrow \infty} \frac{\left\lfloor\frac{a^{2}}{2}\right\rfloor}{\left\lceil\frac{a^{2}}{2}\right\rceil}.$$
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