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Chapter 2: Problem 4

Prove that the following sequence is convergent: \(a_{n}=1+\frac{1}{2 !}+\frac{1}{3 !}+\cdots+\frac{1}{n !}, n \in \mathbb{N} .\)

Short Answer

Expert verified
The sequence \(a_n\) is convergent because it is bounded and monotonic.

Step by step solution

01

Write the Sequence Formally

The sequence is given by \(a_n = 1 + \frac{1}{2!} + \frac{1}{3!} + \cdots + \frac{1}{n!}\). Express it mathematically as \( a_n = \sum_{k=2}^{n} \frac{1}{k!} + 1\).
02

Determine if the Sequence is Bounded

The terms of the sequence, starting from 1, are all positive and are of the form \(\frac{1}{k!}\). These terms decrease very rapidly because the factorial function in the denominator grows very fast. Thus, the sequence is bounded below by 1 since adding positive terms, which approach zero, can't make it less than 1.
03

Consider the Convergence of the Partial Sums

Notice that as \(n\) increases, each individual term \(\frac{1}{n!}\) becomes smaller and approaches 0 due to the factorial function in the denominator. Consequently, the series itself approaches a finite sum due to the decrease in the terms' magnitude.
04

Use Monotonicity of the Sequence

Each additional term \(\frac{1}{n!}\) is positive, making each \(a_n\) greater than the previous term \(a_{n-1}\). Thus, the sequence is non-decreasing.
05

Apply the Monotone Convergence Theorem

Because \(a_n\) is a non-decreasing and bounded sequence (bounded below by 1 and approaches a limit), by the Monotone Convergence Theorem, the sequence \(a_n\) is convergent. Therefore, \( \lim_{n \to \infty} a_n \) exists and is finite.

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Key Concepts

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

Factorial Function
The factorial function, denoted by an exclamation mark (!), is a mathematical concept that multiplies a series of descending natural numbers. For any positive integer , the factorial is calculated as:\[ n! = n \times (n-1) \times (n-2) \times \cdots \times 1 \]This function increases rapidly. For instance, while 3! equals 6, by 5! it reaches 120, and 10! becomes a massive 3,628,800. This rapid growth explains why terms of the form \(\frac{1}{k!}\) in our sequence become very small very quickly.
  • 0! is defined as 1
  • The factorial is central in permutations and combinations.
  • In the sequence problem, \(n!\) helps in understanding how quickly terms approach zero.
By ensuring the denominator grows faster than the numerator, the terms get smaller, influencing convergence.
Bounded Sequences
A sequence is considered bounded if there exists a real number that puts a cap on the values the sequence can take. In simpler terms, a sequence \(a_n\) is bounded if there are numbers \(M\) and \(m\) such that:\[ m \leq a_n \leq M \] for all \(n\).In our problem, the sequence \(a_n = 1 + \frac{1}{2!} + \frac{1}{3!} + \cdots + \frac{1}{n!}\) is bounded below by 1. This is because the first term is 1, and we only add positive fractions which are shrinking as \(n\) grows.
  • Lower Bounds: Place a minimum limit the sequence can鈥檛 go below, like 1 here.
  • Upper Bounds: Hypothetical here, yet bounded by the convergence.
Understanding bounded sequences ensures you know the sequence鈥檚 limits, providing a stepping stone to recognize convergence abilities.
Monotone Sequences
In mathematics, a sequence is called monotone if it is either entirely non-increasing or non-decreasing. In the case of a non-decreasing or an increasing sequence, every term is less than or equal to the next, meaning for any \(n\):\[ a_n \leq a_{n+1} \]For our sequence \(a_n\), it is monotone increasing because every additional term added, \(\frac{1}{n!}\), is positive. This guarantees that each term is at least as large as the previous one.
  • Non-decreasing: Each subsequent term is greater than or equal to the one before, like our sequence.
  • Non-increasing: A decreasing sequence where each term is smaller or equal to the previous.
Recognizing monotonicity aids in applying further theorems, such as the Monotone Convergence Theorem, to determine convergence.
Monotone Convergence Theorem
The Monotone Convergence Theorem is a fundamental principle in analysis, crucial for proving the convergence of sequences. It states that if a sequence is both monotone (non-decreasing or non-increasing) and bounded, then it must converge. In examining our sequence \(a_n\), which is monotone increasing and bounded below, the theorem guarantees that it will converge to a certain limit as \(n\) approaches infinity.
  • Requirement of Boundedness: There must be a limit to how large or small the sequence can grow.
  • Monotonicity: Ensures a single direction of movement towards convergence.
Applying this theorem simplifies proving sequence limits by ensuring conditions are met, making it indispensable for analysts addressing convergence questions.

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Most popular questions from this chapter

Prove that \(\lim _{n \rightarrow \infty} \frac{\sin n}{n}=0\)

Find the following limits if they exist: (a) \(\lim _{n \rightarrow \infty}\left(\sqrt{n^{2}+n}-n\right)\). (b) \(\lim _{n \rightarrow \infty}\left(\sqrt[3]{n^{3}+3 n^{2}}-n\right)\). (c) \(\lim _{n \rightarrow \infty}\left(\sqrt[3]{n^{3}+3 n^{2}}-\sqrt{n^{2}+n}\right)\). (d) \(\lim _{n \rightarrow \infty}(\sqrt{n+1}-\sqrt{n})\). (e) \(\lim _{n \rightarrow \infty}(\sqrt{n+1}-\sqrt{n}) / n\).

Prove that each of the following sequences is convergent and find its limit. (a) \(a_{1}=1\) and \(a_{n+1}=\frac{a_{n}+3}{2}\) for \(n \geq 1\). (b) \(a_{1}=\sqrt{6}\) and \(a_{n+1}=\sqrt{a_{n}+6}\) for \(n \geq 1\). (c) \(a_{n+1}=\frac{1}{3}\left(2 a_{n}+\frac{1}{a_{n}^{2}}\right), n \geq 1, a_{1}>0\). (d) \(a_{n+1}=\frac{1}{2}\left(a_{n}+\frac{b}{a_{n}}\right), b>0\).

Let \(a_{n} \geq 0\) for all \(n \in \mathbb{N}\). Prove that if \(\lim _{n \rightarrow \infty} a_{n}=\ell\), then \(\lim _{n \rightarrow \infty} \sqrt{a_{n}}=\sqrt{\ell}\).

Prove or disprove the following statements: (a) If \(\left\\{a_{n}\right\\}\) and \(\left\\{b_{n}\right\\}\) are convergent sequences, then \(\left\\{a_{n}+b_{n}\right\\}\) is a convergent sequence. (b) If \(\left\\{a_{n}\right\\}\) and \(\left\\{b_{n}\right\\}\) are divergent sequences, then \(\left\\{a_{n}+b_{n}\right\\}\) is divergent sequence. (c) If \(\left\\{a_{n}\right\\}\) and \(\left\\{b_{n}\right\\}\) are convergent sequences, then \(\left\\{a_{n} b_{n}\right\\}\) is a convergent sequence. (d) If \(\left\\{a_{n}\right\\}\) and \(\left\\{b_{n}\right\\}\) are divergent sequences, then \(\left\\{a_{n} b_{n}\right\\}\) is a divergent sequence. (e) If \(\left\\{a_{n}\right\\}\) and \(\left\\{a_{n}+b_{n}\right\\}\) are convergent sequences, then \(\left\\{b_{n}\right\\}\) is a convergent sequence. (f) If \(\left\\{a_{n}\right\\}\) and \(\left\\{a_{n}+b_{n}\right\\}\) are divergent sequences, then \(\left\\{b_{n}\right\\}\) is a divergent sequence.
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