Series

Series: The Sum of a Sequence

When you have a sequence — an ordered list of numbers — the natural next step is to add its terms together. That sum, written as $a_1 + a_2 + a_3 + \dots + a_n + \dots$, is called a series.

The series inherits its nature from the sequence that generates it. If the sequence is finite, the series is finite; if the sequence goes on forever, the series is infinite. For instance, the finite sequence $1, 3, 5, 7$ gives the finite series $1 + 3 + 5 + 7$, while the infinite sequence $1, 3, 5, 7, \dots$ gives the infinite series $1 + 3 + 5 + 7 + \dots$.

Sigma Notation: A Compact Way to Write Series

Writing out long sums term by term is tedious. Mathematicians use the Greek letter sigma ($\Sigma$) as a shorthand for summation. The series $a_1 + a_2 + a_3 + \dots + a_n$ is written compactly as:

The notation reads: "the sum of $a_k$ as $k$ runs from $1$ to $n$." The variable $k$ is called the index of summation; it starts at the lower limit ($1$) and increases by $1$ each step until it reaches the upper limit ($n$).

Working with Sequences and Series: Examples

The textbook demonstrates how to find terms of a sequence from a given rule, and then how to write the corresponding series. Let's walk through each example carefully.

Example 1: Write the first three terms of the sequences defined by:

1. $a_n = 2n + 5$
2. $a_n = \frac{n-3}{4}$ Solution (i): For $a_n = 2n + 5$:

• $n = 1$: $a_1 = 2(1) + 5 = 7$
• $n = 2$: $a_2 = 2(2) + 5 = 9$
• $n = 3$: $a_3 = 2(3) + 5 = 11$

The first three terms are $7, 9, 11$.

Solution (ii): For $a_n = \frac{n-3}{4}$:

• $n = 1$: $a_1 = \frac{1-3}{4} = \frac{-2}{4} = -\frac{1}{2}$
• $n = 2$: $a_2 = \frac{2-3}{4} = \frac{-1}{4} = -\frac{1}{4}$
• $n = 3$: $a_3 = \frac{3-3}{4} = \frac{0}{4} = 0$

The first three terms are $-\frac{1}{2}, -\frac{1}{4}, 0$.

Example 2: Find the 20th term of the sequence defined by $a_n = (n-1)(2-n)(3+n)$.

Solution: Substitute $n = 20$ directly into the formula:

Now multiply step by step:

$19 \times (-18) = -342$

Then $(-342) \times 23 = -7866$

So $a_{20} = -7866$.

Example 3: A sequence is defined recursively: $a_1 = 1$, and $a_n = a_{n-1} + 2$ for $n \ge 2$. Find the first five terms and write the corresponding series.

Solution: A recursive definition gives each term based on the previous one. We start with $a_1 = 1$.

• $a_2 = a_1 + 2 = 1 + 2 = 3$
• $a_3 = a_2 + 2 = 3 + 2 = 5$
• $a_4 = a_3 + 2 = 5 + 2 = 7$
• $a_5 = a_4 + 2 = 7 + 2 = 9$

The first five terms are $1, 3, 5, 7, 9$. The corresponding series is:

The Fibonacci Sequence: A Special Recursive Definition

The textbook introduces the famous Fibonacci sequence in its exercise. It is defined by:

• $a_1 = 1$, $a_2 = 1$ (two initial terms)
• $a_n = a_{n-1} + a_{n-2}$ for $n > 2$ …