Skills Review for Alternating Series and Ratio and Root Tests

Learning Outcomes

  • Apply factorial notation
  • Simplify expressions using the Product Property of Exponents
  • Calculate the limit of a function as đť‘Ą increases or decreases without bound
  • Recognize when to apply L’HĂ´pital’s rule

In the Alternating Series and Ratio and Root Tests sections, we will learn about the last few methods that can be used to determine whether an infinite series diverges or converges. Here we will review how to use factorial notation, use product rule for exponents, take limits at infinity, and apply L’Hopital’s Rule.

Apply Factorial Notation

Recall that [latex]n[/latex] factorial, written as [latex]n![/latex], is the product of the positive integers from 1 to [latex]n[/latex]. For example,

[latex]\begin{align}4!&=4\cdot 3\cdot 2\cdot 1=24 \\ 5!&=5\cdot 4\cdot 3\cdot 2\cdot 1=120\\ \text{ } \end{align}[/latex]

An example of formula containing a factorial is [latex]{a}_{n}=\left(n+1\right)![/latex]. The sixth term of the sequence can be found by substituting 6 for [latex]n[/latex].

[latex]\begin{align}{a}_{6}=\left(6+1\right)!=7!=7\cdot 6\cdot 5\cdot 4\cdot 3\cdot 2\cdot 1=5040 \\ \text{ }\end{align}[/latex]

The factorial of any whole number [latex]n[/latex] is [latex]n\left(n - 1\right)![/latex] We can therefore also think of [latex]5![/latex] as [latex]5\cdot 4!\text{.}[/latex]

A GENERAL NOTE: FACTORIAL

n factorial is a mathematical operation that can be defined using a recursive formula. The factorial of [latex]n[/latex], denoted [latex]n![/latex], is defined for a positive integer [latex]n[/latex] as:

[latex]\begin{array}{l}0!=1\\ 1!=1\\ n!=n\left(n - 1\right)\left(n - 2\right)\cdots \left(2\right)\left(1\right)\text{, for }n\ge 2\end{array}[/latex]

The special case [latex]0![/latex] is defined as [latex]0!=1[/latex].

Try It

Try It

Expand [latex](n+3)![/latex].

Use the Product Rule for Exponents

A General Note: The Product Rule of Exponents

For any real number [latex]a[/latex] and natural numbers [latex]m[/latex] and [latex]n[/latex], the product rule of exponents states that

[latex]{a}^{m}\cdot {a}^{n}={a}^{m+n}[/latex]

Example: Using the Product Rule

Write each of the following products with a single base. Do not simplify further.

  1. [latex]{t}^{5}\cdot {t}^{3}[/latex]
  2. [latex]\left(-3\right)^{5}\cdot \left(-3\right)[/latex]
  3. [latex]{x}^{2}\cdot {x}^{5}\cdot {x}^{3}[/latex]

Try It

RECALL

For any real number [latex]a[/latex] and positive integers [latex]m[/latex] and [latex]n[/latex], the power rule of exponents states that

[latex]{\left({a}^{m}\right)}^{n}={a}^{m\cdot n}[/latex]

 

For an expression like [latex](a^2)^3[/latex], you have a base of [latex]a[/latex] raised to the power of [latex]2[/latex], which is then raised to another power of [latex]3[/latex]. Multiply the exponents [latex]2[/latex] and [latex]3[/latex] to find the new exponent for [latex]a[/latex]. This gives you [latex]a^{2\cdot3}[/latex] or [latex]a^6[/latex]. Always remember: when an exponent is raised to another exponent, multiply the exponents to simplify the expression.

Try It

Simplify the expression [latex](3a^2b)^3 \cdot (2ab^4)[/latex].

Try It

Simplify the expression [latex](2y^2)^3 \cdot (4y^5).[/latex]

Take Limits at Infinity

(see Module 5, Skills Review for Sequences.)

Infinite Limits at Infinity

(see Module 5, Skills Review for Sequences.)

Apply L’HĂ´pital’s Rule

(see Module 5, Skills Review for Sequences.)