4.7 Sigma Notation and Summation | Sequence And Series | Mathematics 11

Sigma Notation

Sigma notation (summation notation) provides a compact way to write the sum of a sequence of terms. The Greek capital letter $Σ$ (sigma) is used:

$\sum_{k = 1}^{n} a_{k} = a_{1} + a_{2} + a_{3} + \dots + a_{n}$

Here:

$k$ is the index of summation (dummy variable)
$1$ is the lower limit
$n$ is the upper limit
$a_{k}$ is the general term

Examples of Sigma Notation

Expanded Form	Sigma Form
$1 + 2 + 3 + \dots + n$	$k = 1 \sum n k$
$1^{2} + 2^{2} + 3^{2} + \dots + n^{2}$	$k = 1 \sum n k^{2}$
$a + a r + a r^{2} + \dots + a r^{n - 1}$	$k = 0 \sum n - 1 a r^{k}$

Arithmetic-Geometric Sequence (AGS)

An arithmetic-geometric sequence is formed by multiplying corresponding terms of an arithmetic sequence and a geometric sequence.

If ${a_{k}}$ is an arithmetic sequence with first term $a$ and common difference $d$ , and ${r^{k - 1}}$ is a geometric sequence with common ratio $r$ , then the arithmetic-geometric sequence is:

$a, (a + d) r, (a + 2 d) r^{2}, (a + 3 d) r^{3}, \dots$

General Term

The $n$ -th term (general term) of an arithmetic-geometric sequence is:

$T_{n} = [a + (n - 1) d] r^{n - 1}$

where:

$a$ = first term of the arithmetic part
$d$ = common difference
$r$ = common ratio of the geometric part

Sum to $n$ Terms

Let $S_{n} = k = 1 \sum n [a + (k - 1) d] r^{k - 1}$ .

Using the multiply-and-subtract method:

$S_{n} = a + (a + d) r + (a + 2 d) r^{2} + \dots + [a + (n - 1) d] r^{n - 1}$

Multiply both sides by $r$ :

$r S_{n} = a r + (a + d) r^{2} + (a + 2 d) r^{3} + \dots + [a + (n - 1) d] r^{n}$

Subtracting:

$(1 - r) S_{n} = a + d r + d r^{2} + \dots + d r^{n - 1} - [a + (n - 1) d] r^{n}$

$(1 - r) S_{n} = a + d \cdot \frac{r ( 1 - r ^{n - 1} )}{1 - r} - [a + (n - 1) d] r^{n}$

$S_{n} = \frac{a}{1 - r} + \frac{d r ( 1 - r ^{n - 1} )}{( 1 - r ) ^{2}} - \frac{[ a + ( n - 1 ) d ] r ^{n}}{1 - r}, r \neq = 1$

Sum to Infinity

When $∣ r ∣ < 1$ , as $n \to \infty$ , the terms $r^{n} \to 0$ and $r^{n - 1} \to 0$ , so:

$S_{\infty} = \frac{a}{1 - r} + \frac{d r}{( 1 - r ) ^{2}}, ∣ r ∣ < 1$

Note: If $∣ r ∣ \geq 1$ , the series diverges and has no finite sum.

Worked Example

Find the sum to infinity of the arithmetic-geometric series: $1 + 2 \cdot \frac{1}{2} + 3 \cdot (\frac{1}{2})^{2} + 4 \cdot (\frac{1}{2})^{3} + \dots$

Here $a = 1$ , $d = 1$ , $r = \frac{1}{2}$ .

Since $∣ r ∣ = \frac{1}{2} < 1$ :

$S_{\infty} = \frac{1}{1 - \frac{1}{2}} + \frac{1 \cdot \frac{1}{2}}{( 1 - \frac{1}{2} ) ^{2}} = \frac{1}{\frac{1}{2}} + \frac{\frac{1}{2}}{\frac{1}{4}} = 2 + 2 = 4$

Standard Summation Formulas (Sigma)

$\sum_{k = 1}^{n} k = \frac{n ( n + 1 )}{2}$

$\sum_{k = 1}^{n} k^{2} = \frac{n ( n + 1 ) ( 2 n + 1 )}{6}$

$\sum_{k = 1}^{n} k^{3} = [\frac{n ( n + 1 )}{2}]^{2}$

$\sum_{k = 1}^{n} c = c n (c is a constant)$

Sigma Notation

Sigma notation (summation notation) provides a compact way to write the sum of a sequence of terms. The Greek capital letter $Σ$ (sigma) is used:

$\sum_{k = 1}^{n} a_{k} = a_{1} + a_{2} + a_{3} + \dots + a_{n}$

Here:

$k$ is the index of summation (dummy variable)
$1$ is the lower limit
$n$ is the upper limit
$a_{k}$ is the general term

Examples of Sigma Notation

Expanded Form	Sigma Form
$1 + 2 + 3 + \dots + n$	$k = 1 \sum n k$
$1^{2} + 2^{2} + 3^{2} + \dots + n^{2}$	$k = 1 \sum n k^{2}$
$a + a r + a r^{2} + \dots + a r^{n - 1}$	$k = 0 \sum n - 1 a r^{k}$

Arithmetic-Geometric Sequence (AGS)

An arithmetic-geometric sequence is formed by multiplying corresponding terms of an arithmetic sequence and a geometric sequence.

If ${a_{k}}$ is an arithmetic sequence with first term $a$ and common difference $d$ , and ${r^{k - 1}}$ is a geometric sequence with common ratio $r$ , then the arithmetic-geometric sequence is:

$a, (a + d) r, (a + 2 d) r^{2}, (a + 3 d) r^{3}, \dots$

General Term

The $n$ -th term (general term) of an arithmetic-geometric sequence is:

$T_{n} = [a + (n - 1) d] r^{n - 1}$

where:

$a$ = first term of the arithmetic part
$d$ = common difference
$r$ = common ratio of the geometric part

Sum to $n$ Terms

Let $S_{n} = k = 1 \sum n [a + (k - 1) d] r^{k - 1}$ .

Using the multiply-and-subtract method:

$S_{n} = a + (a + d) r + (a + 2 d) r^{2} + \dots + [a + (n - 1) d] r^{n - 1}$

Multiply both sides by $r$ :

$r S_{n} = a r + (a + d) r^{2} + (a + 2 d) r^{3} + \dots + [a + (n - 1) d] r^{n}$

Subtracting:

$(1 - r) S_{n} = a + d r + d r^{2} + \dots + d r^{n - 1} - [a + (n - 1) d] r^{n}$

$(1 - r) S_{n} = a + d \cdot \frac{r ( 1 - r ^{n - 1} )}{1 - r} - [a + (n - 1) d] r^{n}$

$S_{n} = \frac{a}{1 - r} + \frac{d r ( 1 - r ^{n - 1} )}{( 1 - r ) ^{2}} - \frac{[ a + ( n - 1 ) d ] r ^{n}}{1 - r}, r \neq = 1$

Sum to Infinity

When $∣ r ∣ < 1$ , as $n \to \infty$ , the terms $r^{n} \to 0$ and $r^{n - 1} \to 0$ , so:

$S_{\infty} = \frac{a}{1 - r} + \frac{d r}{( 1 - r ) ^{2}}, ∣ r ∣ < 1$

Note: If $∣ r ∣ \geq 1$ , the series diverges and has no finite sum.

Worked Example

Find the sum to infinity of the arithmetic-geometric series: $1 + 2 \cdot \frac{1}{2} + 3 \cdot (\frac{1}{2})^{2} + 4 \cdot (\frac{1}{2})^{3} + \dots$

Here $a = 1$ , $d = 1$ , $r = \frac{1}{2}$ .

Since $∣ r ∣ = \frac{1}{2} < 1$ :

$S_{\infty} = \frac{1}{1 - \frac{1}{2}} + \frac{1 \cdot \frac{1}{2}}{( 1 - \frac{1}{2} ) ^{2}} = \frac{1}{\frac{1}{2}} + \frac{\frac{1}{2}}{\frac{1}{4}} = 2 + 2 = 4$

Standard Summation Formulas (Sigma)

$\sum_{k = 1}^{n} k = \frac{n ( n + 1 )}{2}$

$\sum_{k = 1}^{n} k^{2} = \frac{n ( n + 1 ) ( 2 n + 1 )}{6}$

$\sum_{k = 1}^{n} k^{3} = [\frac{n ( n + 1 )}{2}]^{2}$

$\sum_{k = 1}^{n} c = c n (c is a constant)$