Problem 21
Question
In Exercises 21-24, simplify the ratio of factorials. \(\frac{(n+1) !}{n !}\)
Step-by-Step Solution
Verified Answer
The simplified version of the given ratio is n+1.
1Step 1: Understand factorials
A factorial, denoted by '!', is the product of all positive integers from 1 to a given number. So, if we have n!, it means we have the product of all integers from 1 to n. Similarly, (n+1)! would imply the product of all integers from 1 to n, and then multiplied by (n+1).
2Step 2: Expand the factorials.
If we rewrite our ratio by expanding the factorials, we have \(\frac{(n+1)(n)(n-1)(n-2)...(3)(2)(1)}{(n)(n-1)(n-2)...(3)(2)(1)}\). We can notice that the numerator includes all terms from the denominator and an additional (n+1) term.
3Step 3: Simplify the ratio.
Since every term in the denominator is also in the numerator, they cancel each other out leaving \( \frac{n+1}{1}\), which simplifies to n+1.
Key Concepts
FactorialsSimplifying ExpressionsMathematical Reasoning
Factorials
Factorials are a fundamental concept in mathematics, often denoted with an exclamation point '!'. The factorial of a non-negative integer n, written as n!, is the product of all positive integers less than or equal to n. For example, the factorial of 4, which is 4!, would be calculated as:
4! = 4 × 3 × 2 × 1 = 24.
Factorials grow extremely rapidly with increasing values of n, which is why they're frequently found in the fields of combinatorics, algebra, and mathematical analysis.
When simplifying ratios of factorials, such as the exercise presents ------------------------------------------------------------------------------------------------------------ , it's important to recognize that factorials can be expanded and then reduced by cancelling out common factors from the numerator and denominator. One doesn't usually calculate factorials explicitly before simplifying, because this can become impractical for even moderately large n. Instead, mathematical reasoning is applied to identify and eliminate common terms.
4! = 4 × 3 × 2 × 1 = 24.
Factorials grow extremely rapidly with increasing values of n, which is why they're frequently found in the fields of combinatorics, algebra, and mathematical analysis.
When simplifying ratios of factorials, such as the exercise presents ------------------------------------------------------------------------------------------------------------ , it's important to recognize that factorials can be expanded and then reduced by cancelling out common factors from the numerator and denominator. One doesn't usually calculate factorials explicitly before simplifying, because this can become impractical for even moderately large n. Instead, mathematical reasoning is applied to identify and eliminate common terms.
Simplifying Expressions
Simplification in mathematics refers to the process of reducing complexity while maintaining the equivalence of an expression. This process is instrumental in solving equations, evaluating expressions, and can simplify complex problems into more manageable ones.
In our ratio of factorials scenario, simplification leads to recognizing that the numerator includes the factorial of n multiplied by (n+1). Since n! appears in both the numerator and denominator, these extensive products cancel out, except for the (n+1) term that is left over, leading to the final simplified result. This technique showcases the effectiveness of simplification, turning what may seem like a daunting ratio into a simple arithmetic expression.
Techniques for Simplification
Simplifying expressions often involve strategies such as combining like terms, factoring, expanding expressions, and cancelling out terms. In the case of factorials:- Identify common terms in the numerator and denominator.
- Cancel out the identical factors.
- Rewrite the expression without the cancelled terms to find the simplified form.
In our ratio of factorials scenario, simplification leads to recognizing that the numerator includes the factorial of n multiplied by (n+1). Since n! appears in both the numerator and denominator, these extensive products cancel out, except for the (n+1) term that is left over, leading to the final simplified result. This technique showcases the effectiveness of simplification, turning what may seem like a daunting ratio into a simple arithmetic expression.
Mathematical Reasoning
Mathematical reasoning involves the process of thinking logically and creating a coherent argument to solve mathematical problems. This skill is essential for identifying patterns, formulating conjectures, and proving or disproving statements mathematically.
In the given exercise, mathematical reasoning is utilized to understand what factorials represent and how the properties of factorials can be strategically used to simplify expressions. By recognizing that the numerator contains an extra (n+1) factor that is not in the denominator, we logically deduce that all other factors cancel out. This conclusion does not necessarily require computation but rather a logical understanding of the relationships between these mathematical elements. As a result, we reach a simplified answer with minimal calculation, demonstrating the power and elegance of mathematical reasoning.
Logical Steps in Mathematical Reasoning
To apply mathematical reasoning:- Understand the problem and the mathematical concepts involved.
- Identify relationships and patterns that can simplify the problem.
- Apply knowledge of mathematical properties and operations logically.
- Proceed step by step to ensure each stage of solution is valid.
In the given exercise, mathematical reasoning is utilized to understand what factorials represent and how the properties of factorials can be strategically used to simplify expressions. By recognizing that the numerator contains an extra (n+1) factor that is not in the denominator, we logically deduce that all other factors cancel out. This conclusion does not necessarily require computation but rather a logical understanding of the relationships between these mathematical elements. As a result, we reach a simplified answer with minimal calculation, demonstrating the power and elegance of mathematical reasoning.
Other exercises in this chapter
Problem 20
Use the power series $$\frac{1}{1+x}=\sum_{n=0}^{\infty}(-1)^{n} x^{n}$$ to determine a power series, centered at 0 , for the function. Identify the interval of
View solution Problem 21
Verify that the infinite series converges. $$ \sum_{n=0}^{\infty} 2\left(\frac{3}{4}\right)^{n} $$
View solution Problem 21
In Exercises \(19-24,\) find the \(n\) th Taylor polynomial centered at \(c\). $$ f(x)=\sqrt{x}, \quad n=4, \quad c=1 $$
View solution Problem 21
In Exercises \(21-30,\) find the Maclaurin series for the function. (Use the table of power series for elementary functions.) $$ f(x)=e^{x^{2} / 2} $$
View solution