Problem 60
Question
Write an expression for the \(n\) th term of the sequence. (There is more than one correct answer.) \(1, \frac{1}{2}, \frac{1}{6}, \frac{1}{24}, \frac{1}{120}, \ldots\)
Step-by-Step Solution
Verified Answer
The \(n^{th}\) term of the sequence can be expressed as \(\frac{1}{n!}\).
1Step 1: Identifying the pattern
Analyzing the sequence, it seems that each term is of the form \(\frac{a}{n!}\). This formulates the hypothesis that the sequence is related to factorials. For instance, \(1=1!, \frac{1}{2} = \frac{1}{2!}, \frac{1}{6} = \frac{1}{3!}, \frac{1}{24} = \frac{1}{4!}, \frac{1}{120} = \frac{1}{5!}\). The relationship seems to be \(an = \frac{1}{n!}\).
2Step 2: Formulating the nth term
We can express the \(n^{th}\) term as \(a_n = \frac{1}{n!}\). The term \(a_n\) represents the \(n^{th}\) term in the sequence. Here, '!' indicates a factorial, meaning the product of all positive integers up to that number. For instance, \(3! = 3*2*1 = 6\). This formula gives a method to determine any term in the sequence without knowing its previous terms.
3Step 3: Verifying the formula
Just to be sure of the formula, let's validate this by checking with an item in the sequence, say the fifth term. Substituting \(n = 5\) in \(a_n = \frac{1}{n!}\), we get \(a_5 = \frac{1}{5!} = \frac{1}{120}\). This matches the fifth term in the given sequence. Hence, the formula holds true for the sequence.
Key Concepts
FactorialsSequence PatternsNth Term Formula
Factorials
Factorials are a mathematical concept represented by the symbol '!'. They are the product of all positive integers up to a specified number. For example, the factorial of 3, written as \(3!\), is calculated as \(3 \times 2 \times 1 = 6\). Factorials grow very rapidly, which is why they're useful in many mathematical fields, including combinatorics and probability.
They help to represent sequences, like the one in our exercise, where each term is a fraction with a factorial in the denominator.
They help to represent sequences, like the one in our exercise, where each term is a fraction with a factorial in the denominator.
Sequence Patterns
Understanding sequence patterns is key to identifying how sequences progress. In our example, the given sequence is \(1, \frac{1}{2}, \frac{1}{6}, \frac{1}{24}, \frac{1}{120}, \ldots\). Each term is progressively smaller, representing reciprocals of factorials.
In mathematical terms, a sequence is a set of numbers arranged in a specific order. Patterns can be arithmetic, geometric, or even based on factorials as seen here. Recognizing these patterns helps in predicting future terms.
In mathematical terms, a sequence is a set of numbers arranged in a specific order. Patterns can be arithmetic, geometric, or even based on factorials as seen here. Recognizing these patterns helps in predicting future terms.
Nth Term Formula
The nth term formula of a sequence allows us to find any term without listing all the previous terms. For factorial-based sequences, it is often written as \(a_n = \frac{1}{n!}\).
The formula provides a straightforward calculation for any specified term. For instance, to find the 5th term, we substitute \(n = 5\) into the formula, giving \(a_5 = \frac{1}{5!} = \frac{1}{120}\).
The formula provides a straightforward calculation for any specified term. For instance, to find the 5th term, we substitute \(n = 5\) into the formula, giving \(a_5 = \frac{1}{5!} = \frac{1}{120}\).
- This saves time and effort, especially for large sequences.
- It is a universal approach applicable to any factorial sequence.
Other exercises in this chapter
Problem 59
In Exercises \(59-62,\) verify that the Ratio Test is inconclusive for the \(p\) -series. $$ \sum_{n=1}^{\infty} \frac{1}{n^{3 / 2}} $$
View solution Problem 60
Determine the convergence or divergence of the series. $$ \sum_{n=0}^{\infty} \frac{1}{4^{n}} $$
View solution Problem 60
Let \(f(x)=\sum_{n=0}^{\infty} \frac{x^{n}}{n !}\) (a) Find the interval of convergence of \(f\). (b) Show that \(f^{\prime}(x)=f(x)\). (c) Show that \(f(0)=1\)
View solution Problem 60
Use the Limit Comparison Test to determine the convergence or divergence of the series. $$ \sum_{n=1}^{\infty} \frac{5}{n+\sqrt{n^{2}+4}} $$
View solution