Sequences and Series

Hong Kong

Stage 4 - Stage 5

Lesson

Writing out the partial sum of a sequence can take up a lot of space and time. A number of centuries ago, mathematicians succeeded in developing a short hand notation for many sequences, and they borrowed from the Greek alphabet to do it.

The symbol $\Sigma$Σ (pronounced "sigma") is the capital letter S in the Greek alphabet. When $\Sigma$Σ is used to express a series in mathematics, then in those instances it stands for the word "Sum". To explain how the symbol is used, consider the following expression:

$\sum_{n=1}^{n=5}$`n`=5∑`n`=1 $n^2$`n`2

The equation $n=1$`n`=1 directly below and to the right of the $\Sigma$Σ sign tells us that we start the series by substituting $n=1$`n`=1 into the sequence formula shown as $n^2$`n`2. So the series begins:

$1^2$12

Then we increase $n$`n` by $1$1 so that $n$`n` becomes $2$2. The new value of $n$`n` is substituted into the sequence formula to reveal $2^2$22 so that the series becomes:

$1^2+2^2$12+22

Again we increase $n$`n` by $1$1 and substitute for the third term so that the series becomes:

$1^2+2^2+3^2$12+22+32

This process of increasing $n$`n` by $1$1 continues until $n=5$`n`=5 is reached as shown directly above and to the right of the $\Sigma$Σ sign. The complete sum becomes:

$\sum_{n=1}^{n=5}n^2=1^2+2^2+3^2+4^2+5^2$`n`=5∑`n`=1`n`2=12+22+32+42+52

Thus this series sums to $55$55.

When it is clear that the variable being referred to is $n$`n`, we are allowed to drop the "$n$`n`" in the superscripted and subscripted expression so that the above example can be written:

$\sum_1^5n^2=1^2+2^2+3^2+4^2+5^2$5∑1`n`2=12+22+32+42+52

As a second example, the first $100$100 terms of the arithmetic series whose first term is $a=10$`a`=10 and whose common difference is $d=3$`d`=3 is given, with our new notation, as:

$\sum_1^{100}\left(7+3n\right)$100∑1(7+3`n`)

Following the same strategy, you should be able to see that the series written out would look like:

$10+13+16+...+304+307$10+13+16+...+304+307

Note that the common difference, the first term and last term are easy to spot, and this means we can use the sum formula for an arithmetic progression to show that:

$S_{100}=\frac{100}{2}\left(10+307\right)=15850$`S`100=1002(10+307)=15850

In some instances of geometric series, we can add up an infinite number of terms and obtain a finite sum. Specifically this happens when the common ratio $r$`r` is between, but not including, $-1$−1 and $1$1.

Take for example the expression:

$\sum_{n=1}^{\infty}32\times\left(\frac{1}{2}\right)^{n-1}$∞∑`n`=132×(12)`n`−1

Using our strategy, we can write the first few terms as $32+16+8+4+...$32+16+8+4+... and using the limiting sum formula, we find that the finite sum is given by:

$S_{\infty}=\frac{a}{1-r}=\frac{32}{1-\left(\frac{1}{2}\right)}=64$`S`∞=`a`1−`r`=321−(12)=64

As a final note, the summation notation is quite versatile in its application to series. For example, the mean of a set of scores, say in general terms the scores $x_1,x_2,x_3,...,x_n$`x`1,`x`2,`x`3,...,`x``n` is their sum divided by $n$`n`.

We can write this using what is known as a dummy variable $i$`i` and write:

$\sum_{i=1}^n$`n`∑`i`=1 $\frac{x_i}{n}$`x``i``n`

In expanded form this expression reads:

$\frac{x_1}{n}+\frac{x_2}{n}+\frac{x_3}{n}+...+\frac{x_n}{n}$`x`1`n`+`x`2`n`+`x`3`n`+...+`x``n``n`

or equivalently $\frac{x_1+x_2+x_3+...+x_n}{n}$`x`1+`x`2+`x`3+...+`x``n``n`.

Write the following series using sigma notation.

$4+8+12+16+20+\text{. . .}$4+8+12+16+20+. . .

$\sum_{k=1}^{\infty}\left(\editable{}\right)$∞∑

`k`=1()

Consider the series:

$\frac{1}{1^3}+\frac{1}{2^3}+\frac{1}{3^3}+\frac{1}{4^3}+\frac{1}{5^3}$113+123+133+143+153

Rewrite the series using sigma notation in the form $\sum_{k=\editable{}}^{\editable{}}\editable{}$∑`k`=.

Find the value of $\sum_{r=1}^4\frac{1}{r+2}$4∑`r`=11`r`+2.