Skills Review for Power Series and Functions

Learning Outcomes

  • Use summation notation
  • Apply factorial notation
  • Simplify expressions using the Product Property of Exponents

In the Power Series and Functions section, we will look at power series and how they basically represent infinite polynomials. Here we will review how to expand sigma (summation) notation, apply factorial notation, and use the product rule for exponents.

Expand Sigma (Summation) Notation

(also in Module 1, Skills Review for Approximating Areas)

Summation notation is used to represent long sums of values in a compact form. Summation notation is often known as sigma notation because it uses the Greek capital letter sigma to represent the sum. Summation notation includes an explicit formula and specifies the first and last terms of the sum. An explicit formula for each term of the series is given to the right of the sigma. A variable called the index of summation is written below the sigma. The index of summation is set equal to the lower limit of summation, which is the number used to generate the first term of the sum. The number above the sigma, called the upper limit of summation, is the number used to generate the last term of the sum.

If we interpret the given notation, we see that it asks us to find the sum of the terms in the series [latex]{a}_{i}=2i[/latex] for [latex]i=1[/latex] through [latex]i=5[/latex]. We can begin by substituting the terms for [latex]i[/latex] and listing out the terms.

[latex]\begin{array}{l} {a}_{1}=2\left(1\right)=2 \\ {a}_{2}=2\left(2\right)=4\hfill \\ {a}_{3}=2\left(3\right)=6\hfill \\ {a}_{4}=2\left(4\right)=8\hfill \\ {a}_{5}=2\left(5\right)=10\hfill \end{array}[/latex]

We can find the sum by adding the terms:

[latex]\displaystyle\sum _{i=1}^{5}2i=2+4+6+8+10=30[/latex]

A General Note: Summation Notation

The sum of the first [latex]n[/latex] terms of a series can be expressed in summation notation as follows:

[latex]\displaystyle\sum _{i=1}^{n}{a}_{i}[/latex]

This notation tells us to find the sum of [latex]{a}_{i}[/latex] from [latex]i=1[/latex] to [latex]i=n[/latex].

[latex]k[/latex] is called the index of summation, 1 is the lower limit of summation, and [latex]n[/latex] is the upper limit of summation.

Example: EXpanding Summation Notation

Evaluate [latex]\displaystyle\sum _{i=3}^{7}{i}^{2}[/latex].

Try It

Evaluate [latex]\displaystyle\sum _{i=2}^{5}\left(3i - 1\right)[/latex].

 

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Apply Factorial Notation

(also in Module 5, Skills Review for Alternating Series and Ratio and Root Tests)

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

(also in Module 5, Skills Review for Alternating Series and Ratio and Root Tests)

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].

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