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Chapter 9: Problem 75

Prove that \(\lim _{n \rightarrow \infty} \frac{x^{n}}{n !}=0\) for any real \(x\).

Short Answer

Expert verified
Proof for \(\lim _{n \rightarrow \infty} \frac{x^{n}}{n !}=0\) for any real number x has been established by breaking down the sequence, evaluating each term and demonstrating that as n tends to infinity, each term in the sequence will become less than 1 and the whole sequence will tend to 0.

Step by step solution

01

Write down the sequence

Execute this by writing the sequence: \(\frac{x^n}{n!}\), where x is the real number and n is a positive integer that reaches infinity.
02

Break down the sequence

This sequence can be expanded and written as:\(x*\frac{x}{2}*\frac{x}{3}*...*\frac{x}{(n-1)}*\frac{x}{n}\). The number x is divided by each integer from 1 to n.
03

Evaluate the sequence

As n gets larger (tends to infinity), we will eventually reach a point where \(x/i\) (for some i in the sequence) will be less than 1. All subsequent terms will be even smaller since we will be dividing x by even larger numbers. Therefore, the product (and hence the whole sequence) will tend to 0.

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Key Concepts

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

Infinite Sequences
When we talk about infinite sequences in mathematics, we're referring to an ordered list of numbers that goes on indefinitely. For instance, the sequence of all positive integers (1, 2, 3, ...) is an example of such a sequence. In calculus, we often explore what happens to the terms of an infinite sequence as the position in the sequence, typically denoted by n, becomes very large鈥攊n other words, as n approaches infinity.

Understanding how these sequences behave can be critical in many areas of mathematics, including series and differential equations. A particularly intriguing aspect to explore is whether the terms of a sequence get closer to a certain value, which brings us to the concept of 'convergence.' In the context of the exercise provided, examining the behavior of the sequence as n goes to infinity is crucial for determining the limit.
Factorial
The concept of factorial, represented by n!, is fundamental in combinatorics, calculus, and many areas of mathematics. To define it simply, the factorial of a non-negative integer n is the product of all positive integers less than or equal to n.

Mathematically, the factorial of n (!) is defined as:
! = n 脳 (n-1) 脳 (n-2) 脳 ... 脳 3 脳 2 脳 1
For example, 5! = 5 脳 4 脳 3 脳 2 脳 1 = 120.
The factorial function grows very rapidly with the increase of n. This rapid growth is precisely what contributes to the sequence of the factorial becomes very large and aids in causing the fraction to approach 0.
Convergence of Sequences
The concept of the convergence of sequences is a cornerstone of calculus. When we say a sequence converges, we mean that as n increases without bound (as n goes to infinity), the terms of the sequence get closer and closer to a specific number, called the limit.

This is a powerful concept because it helps us understand the behavior of sequences in the long run. If a sequence converges to a limit L, then for any degree of closeness desired, there exists some point in the sequence after which all terms are within that desired distance from L. For the sequence discussed in our exercise , that desired limit happens to be 0. The proof of convergence relies on recognizing that, due to the factorial in the denominator, our fraction will decrease to the point that its value is as close to 0 as we want, for sufficiently large n, indicating the sequence converges to 0.

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Most popular questions from this chapter

Find all values of \(x\) for which the series converges. For these values of \(x\), write the sum of the series as a function of \(x\). $$ \sum_{n=0}^{\infty}(-1)^{n} x^{n} $$

Use the formula for the \(n\) th partial sum of a geometric series \(\sum_{i=0}^{n-1} a r^{i}=\frac{a\left(1-r^{n}\right)}{1-r}\) Present Value The winner of a \(\$ 1,000,000\) sweepstakes will be paid \(\$ 50,000\) per year for 20 years. The money earns \(6 \%\) interest per year. The present value of the winnings is \(\sum_{n=1}^{20} 50,000\left(\frac{1}{1.06}\right)^{n} .\) Compute the present value and interpret its meaning.

Consider the sequence \(\left\\{a_{n}\right\\}=\left\\{\frac{1}{n} \sum_{k=1}^{n} \frac{1}{1+(k / n)}\right\\}\). (a) Write the first five terms of \(\left\\{a_{n}\right\\}\). (b) Show that \(\lim _{n \rightarrow \infty} a_{n}=\ln 2\) by interpreting \(a_{n}\) as a Riemann sum of a definite integral.

Let \(\sum a_{n}\) be a convergent series, and let \(R_{N}=a_{N+1}+a_{N+2}+\cdots\) be the remainder of the series after the first \(N\) terms. Prove that \(\lim _{N \rightarrow \infty} R_{N}=0\).

Show that the series \(\sum_{n=1} a_{n}\) can be written in the telescoping form \(\sum_{n=1}^{\infty}\left[\left(c-S_{n-1}\right)-\left(c-S_{n}\right)\right]\) where \(S_{0}=0\) and \(S_{n}\) is the \(n\) th partial sum.
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