# Sequences and Series

#### Sequences

A sequence is an ordered list of numbers such as $a_1, a_2, a_3, \dots$ formed according to a definite rule. Each member in this ordered list is called and "element" or "term" of the sequence. The sequence is defined by the number of terms it contains as either finite or infinite.

The list of positive, even numbers $2, 4, 6, 8, \dots$ is a typical example of an infinite sequence. The dots indicate that the sequence continues forever, with no last term.

The list of positive, even numbers less that $100$ - $2, 4, 6, 8, \dots , 98$ - is an example of a finite sequence. The first term in this sequence is $2$ and the last term is $98$.

##### Arithmetic sequences

An arithmetic sequence is a sequence where each term is obtained from the preceding one by adding a constant amount. This constant is called the common difference and is often represented by the symbol $d$. The value of $d$ can be positive, negative or zero. The sequence is of the form $$a,\; a+d,\; a+2d,\; a+3d, \dots$$ where $a$ is the first term of the sequence. The $n^\text{th}$ term of an arithmetic sequence is given by $$a_n=a+(n-1)d$$.

##### Geometric sequences

A geometric sequence is a sequence where each term is obtained from the preceding one by multiplying by a constant amount. This constant is called a common ratio and is often represented by the symbol $r$. The value of $r$ can be positive or negative ($r=0$ gives a trivial sequence with only one term). The terms of a geometric sequence with negative $r$ will oscillate between positive and negative values. The sequence is of the form $$a,\; ar,\; ar^2,\; ar^3, \dots$$. The $n^\text{th}$ term of a geometric sequence is given by $$a_n=ar^{(n-1)}$$.

#### Series

A series is obtained by adding up the terms of a sequence. If all the terms of a finite sequence are added it is called a finite series. For example:

If $a_1,\; a_2,\; a_3,\dots$ is an infinite sequence , then the corresponding series is $$a_1+a_2+a_3+\cdots$$

The sum of the first $n$ terms of an infinite series is written as $$S_n=a_1+ a_2+ a_3 +\dots +a_n =\sum_{j=1}^na_j$$

##### Arithmetic series

An arithmetic series is a series where the difference between successive terms is a constant. The sum of the first $n$ terms of an arithmetic series with first term $a_1=a$ and last term $a_n=l$ is given by $$S_n=\frac{n}{2}\left( a+l\right).$$

In terms of the common difference between terms, $d$, (see section on sequences above) this can also be written as $$S_n=\frac{n}{2}\left( 2a+(n-1)d\right).$$

##### Geometric series

A geometric series is a series where each term is obtained from the preceding one by multiplying by a constant. The sum of the the first $n$ terms of a geometric series with first term $a_1=a$ and common ratio $=r$ is given by $$S_n=\frac{a(1-r^n)}{1-r},\text{ for }r\ne 1.$$

In the case where $|r|\lt 1$ it can be shown that this sum will tend to $$S_\infty =\frac{a}{1-r}.$$

##### Double series

A double series is a series depending on two indices such as $$\sum_{i,j}a_{i,j}.$$

If the terms of the series can be written as the product of two single subscript series, that is if $a_{i,j}=x_iy_j$, then a finite double series can be written as a product of series. For example: \begin{align*} \sum_{i=1}^m \sum_{j=1}^n x_iy_j&=x_1y_1+x_1y_2+\cdots +x_1y_n+x_2y_1+x_2y_2+\cdots \\ &=(x_1+x_2+\cdots +x_m)y_1+(x_1+x_2+\cdots +x_m)y_2+\cdots \\ &=\left(\sum_{i=1}^m x_i\right)(y_1+y_2+\cdots +y_n) \\ &=\left(\sum_{i=1}^m x_i\right)\left(\sum_{j=1}^n y_j\right) \end{align*}