Skills Review for Series Solutions of Differential Equations

Learning Outcomes

  • Apply factorial notation
  • Use summation notation
  • Write the terms of a sequence defined by a recursive formula

In the Series Solutions of Differential Equations section, we will use infinite series to solve and express solutions of differential equations. Here we will review how to use factorial notation, expand sigma (summation) notation, and evaluate a recursive formula.

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].

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

Expand Sigma (Summation) Notation

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].

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

 

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Evaluate a Recursive Formula

A recursive formula is a formula that defines its value at a particular input using the result of the previous input(s).

A recursive formula always has two parts: the value of an initial input and an equation defining each term in terms of preceding terms. For example, suppose we know the following:

[latex]\begin{align}&{x}_{1}=3 \\ &{x}_{n}=2{x}_{n - 1}-1, \text{ for } n\ge 2 \end{align}[/latex]
We can find the subsequent terms of the recursive formula using the first term.
[latex]\begin{align}&{x}_{1}=3\\ &{x}_{2}=2{x}_{1}-1=2\left(3\right)-1=5\\ &{x}_{3}=2{x}_{2}-1=2\left(5\right)-1=9\\ &{x}_{4}=2{x}_{3}-1=2\left(9\right)-1=17\end{align}[/latex]

So, the first four terms are [latex]3,5,9,\text{ and},17[/latex].

Example: Evaluating a Recursive Formula

Write the first five terms defined by the recursive formula.

[latex]\begin{align}&{x}_{1}=9\\ &{x}_{n}=3{x}_{n - 1}-20\text{, for }n\ge 2\end{align}[/latex]

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