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Chapter 3: Problem 18

Discuss the convergence of the following sequences, where \(a, b\) satisfy \(01\). (a) \(\left(n^{2} a^{n}\right)\), (b) \(\left(b^{n} / n^{2}\right)\) (c) \(\left(b^{n} / n !\right)\), (d) \(\left(n ! / n^{n}\right)\).

Short Answer

Expert verified
All four sequences converge, and they also converge to zero.

Step by step solution

01

Understanding Convergence

A sequence is said to converge if, as the terms go on, they get closer to a certain limit. In other words, we should be able to state a 'limiting' value that the terms of our sequence get arbitrarily close to.
02

Sequence (a)

Consider the sequence \(n^{2} a^{n}\). Here, each term \(n^{2} a^{n}\) becomes small as \(n\) increases (because \(0<a<1\)). Thus, the sequence converges to zero.
03

Sequence (b)

Now consider the sequence \(b^{n} / n^{2}\). While \(b^{n}\) grows logarithmically when \(b>1\), \(n^{2}\) grows quadratically. It is known that polynomial growth is slower than logarithmic growth. Therefore, the sequence will also converge to zero.
04

Sequence (c)

Let's look at the sequence \(b^{n} / n !\). Here, \(n !\), the factorial of \(n\), grows faster than \(b^{n}\). Because of this fact, the sequence \(b^{n} / n !\) also converges to zero.
05

Sequence (d)

Finally, consider the sequence \(n ! / n^{n}\). The term \(n^{n}\) grows significantly faster than \(n !\) as \(n\) increases, leading the terms of the sequence to converge to zero again.

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Key Concepts

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

Limit of a Sequence
Every sequence has a set of terms that follow a specific rule. To determine if a sequence converges, we look for a limit: a value that the terms approach as the sequence progresses. Think of a sequence like a road leading to a final destination, the limit, where terms get closer and closer to that specific point.
In mathematical terms, a sequence \(a_n\) converges to a limit \(L\) if for every small number \(\varepsilon > 0\), there exists a natural number \(N\) such that for all numbers \(|a_n - L| < \varepsilon\) when \(n > N\). Simply put:
  • The difference between terms of the sequence and the limit gets smaller as you go further into the sequence.
  • The sequence "settles" into a particular value, either zero or some finite number.
  • Recognizing convergence helps us understand long-term behavior of mathematical models."
Exponential Growth
Exponential growth happens when the rate of growth of a function is proportional to its current size, meaning it gets faster as time goes on. It is characterized by the expression \(b^n\), where the base \(b\) is greater than 1. As \(n\) increases, \(b^n\) grows incredibly quickly, often outpacing other types of growth like linear or polynomial.
Some key features to remember include:
  • Exponential functions double (or more) every set period of time, which leads to rapid increase.
  • It contrasts with linear or polynomial growth, where it grows much slower.
  • Identifying exponential growth is crucial to predicting patterns in real-world situations, such as population dynamics or compound interest calculations.
Factorial
The factorial of a number, denoted as \(n!\), is a product of all positive integers less than or equal to \(n\). For instance, \(5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\). It represents the number of ways to arrange \(n\) objects, which makes it essential in fields such as combinatorics and probability.
Two key points about factorial growth:
  • Factorial growth is extremely rapid, even faster than exponential growth for large \(n\).
  • In sequences, factorial terms in the denominator usually lead to convergence if paired with an exponential numerator since factorials grow so quickly.
Factorials provide insight into the rapidly increasing number of possible outcomes as parameters expand, such as choosing sets of items or organizing data.
Polynomial Growth
Polynomial growth refers to the increase of a function determined by polynomials, for example, \(n^2\) or \(n^3\). These functions grow at a steady rate characterized by their highest-power term.
Here are some characteristics of polynomial growth:
  • It is slower than exponential growth but faster than linear growth.
  • Polynomial functions are used to describe situations where growth accelerates but not as sharply as exponential functions.
  • Knowing about polynomial growth helps us understand phenomena like the change of certain social trends, physics equations, or computational complexities over time.
In the given sequences, recognizing the difference in growth rates helps you conclude about the sequence's convergence or divergence.

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Most popular questions from this chapter

Let \(y_{n}:=\sqrt{n+1}-\sqrt{n}\) for \(n \in \mathbb{N}\). Show that \(\left(y_{n}\right)\) and \(\left(\sqrt{n} y_{n}\right)\) converge. Find their limits.

Let \(x_{1}>1\) and \(x_{n+1}:=2-1 / x_{n}\) for \(n \in \mathbb{N}\). Show that \(\left(x_{n}\right)\) is bounded and monotone. Find the limir.

Let \(A\) be an infinite subset of \(\mathbb{R}\) that is bounded above and let \(u:=\) sup \(A\). Show there exists an increasing sequence \(\left(x_{n}\right)\) with \(x_{n} \in A\) for all \(n \in \mathbb{N}\) such that \(u=\lim \left(x_{n}\right)\).

Let \(x_{1}:=8\) and \(x_{n+1}:=\frac{1}{2} x_{n}+2\) for \(n \in \mathbb{N}\). Show that \(\left(x_{n}\right)\) is bounded and monotone. Find the limit.

For any \(b \in \mathbb{R}\), prove that \(\lim (b / n)=0\).
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