1.2 Sequences in General – Series

1. Sequence

A sequence is just an ordered list of numbers (terms in a definite order). Usually, the terms of a sequence follow a specific pattern, for example:

  • $0, 2, 4, 6, 8, 10, \dots$ (even numbers)
  • $1, 3, 5, 7, 9, 11, \dots$ (odd numbers)
  • $5, 10, 15, 20, 25, \dots$ (positive multiples of 5)
  • $2, 4, 8, 16, 32, \dots$ (powers of 2)

We use the notation $u_n$ to describe the $n$-th term. Thus, the terms of the sequence are denoted by:

$u_1, u_2, u_3, u_4, u_5, \dots$

2. Series

A series is just a sum of terms. We say that $S_\infty$ is an infinite series, while the finite sums $S_1, S_2, S_3 \dots$ are called partial sums.

$S_n = u_1 + u_2 + u_3 + \dots + u_n$
(the sum of the first $n$ terms)
$S_\infty = u_1 + u_2 + u_3 + \dots$
(the sum of all terms, $\infty$ terms)

EXAMPLE 1

Consider the sequence $1, 3, 5, 7, 9, 11, \dots$ (odd numbers)

Some of the terms are: $u_1=1, u_2=3, u_3=5, u_6=11, u_{10}=19$
Also:
$S_1 = 1$
$S_2 = 1+3 = 4$
$S_3 = 1+3+5 = 9$
$S_4 = 1+3+5+7 = 16$
Finally: $S_\infty = 1+3+5+7+\dots$ (in this case the result is $+\infty$)

3. Sigma Notation $\left(\sum_{n=1}^{k}\right)$

Instead of writing $u_1 + u_2 + u_3 + u_4 + u_5 + u_6 + u_7 + u_8 + u_9$, we may write:

$\sum_{n=1}^{9} u_n$

It stands for the sum of all terms $u_n$, where $n$ ranges from 1 to 9.

In general, $\sum_{n=1}^{k} u_n$ expresses the sum of all terms $u_n$, where $n$ ranges from 1 to $k$. We may also start with another value for $n$, instead of 1, e.g. $\sum_{n=4}^{9} u_n$.

EXAMPLE 2

  • $\sum_{n=1}^{3} 2^n = 2^1 + 2^2 + 2^3 = 2+4+8 = \mathbf{14}$
  • $\sum_{n=1}^{4} \frac{1}{n} = \frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} = \frac{12+6+4+3}{12} = \mathbf{\frac{25}{12}}$
  • $\sum_{k=1}^{3} \frac{1}{2^k} = \frac{1}{2} + \frac{1}{4} + \frac{1}{8} = \frac{4+2+1}{8} = \mathbf{\frac{7}{8}}$
  • $\sum_{n=3}^{6} (2n+1) = 7+9+11+13 = \mathbf{40}$
  • $\sum_{x=3}^{20} (x^2 + x) = (3^2+3) + (4^2+4) + (5^2+5) + \dots = \dots$ whatever that is!

Infinite Sum Example:

  • $\sum_{n=1}^{\infty} \frac{1}{2^n} = \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \dots$ (it never finishes!)

The result is $\mathbf{1}$.

4. Describing a Sequence

There are two basic ways to describe a sequence:

A) By a GENERAL FORMULA

We just describe the general term $u_n$ in terms of $n$.

For example, $u_n = 2n$
It gives $u_1=2, u_2=4, u_3=6, \dots$
It is the sequence $\mathbf{2, 4, 6, 8, 10, \dots}$

EXAMPLE 3

$u_n = n^2$ is the sequence $1^2, 2^2, 3^2, 4^2, 5^2, \dots$
that is $\mathbf{1, 4, 9, 16, 25, \dots}$
$u_n = 2^n$ is the sequence $\mathbf{2, 4, 8, 16, 32, \dots}$

B) By a RECURSIVE RELATION (mainly for Math HL)

Given: $u_1$ (the first term) and $u_{n+1}$ in terms of $u_n$.

For example: $u_1 = 10, \quad u_{n+1} = u_n + 2$

This says that the first term is 10 and then:

  • $u_2 = u_1 + 2$
  • $u_3 = u_2 + 2$
  • $u_4 = u_3 + 2$ and so on.

In simple words, begin with 10 and keep adding 2 in order to find the following term. It is the sequence $\mathbf{10, 12, 14, 16, 18, \dots}$

EXAMPLE 4

$u_1 = 3, \quad u_{n+1} = 2u_n + 5$
It is the sequence $\mathbf{3, 11, 27, 59, \dots}$

EXAMPLE 5

Sometimes, we are given the first two terms $u_1, u_2$ and then a recursive formula for $u_{n+1}$ in terms of $u_n$ and $u_{n-1}$.

The most famous sequence of this form is the Fibonacci sequence:

$u_1 = 1, \quad u_2 = 1$
$u_{n+1} = u_n + u_{n-1}$

In other words, add $u_1, u_2$ in order to obtain $u_3$, add $u_2, u_3$ in order to obtain $u_4$, and so on.

It is the sequence $\mathbf{1, 1, 2, 3, 5, 8, 13, 21, 34, 55, \dots}$