Professor Dave here, let’s talk about sequences.
Let’s count some numbers, shall we?
One.
Two.
Three.
Four.
What are we doing exactly?
We are generating a sequence of numbers, which happens to be the natural numbers.
If a particular term in the series can be represented by A sub N, and the entire sequence
is A sub N in brackets, then this sequence could be represented the letter N, instead
of writing out all the numbers.
This means that A one, the first term of the sequence, when N equals one, is one.
To get A two, the second term, N must equal two, and we get two.
Now let’s list all the even numbers.
Two.
Four.
Six.
Eight.
How would we represent this series?
That would be two N. A one, where N equals one, gives us two.
A two, where N equals two, gives us four, and so forth.
What if we do the natural numbers again but start with two?
Now the first term is two, the second term is three, so we can represent this as N plus
one, where every term is one greater than its chronological number in the sequence.
We can generate sequences using any expression like this, and they can get pretty complex.
Try two N plus three.
The first term would be five.
The second term would be seven.
And then nine, eleven, thirteen.
Sequences like this are called arithmetic sequences, because each term differs from
the previous by a constant amount, in this case by two.
But not all sequences work this way.
There are also geometric sequences, where each term can be attained not by adding a
constant to the previous term, but multiplying the previous term by a constant.
Something like two, six, eighteen, fifty-four, each term is three times the previous.
Can you figure out how to represent this sequence?
It starts with two, and then you multiply by three once more for each term, so this
should be two times three to the N minus one.
Some sequences fall into neither of these categories.
What about two to the N, minus one?
We’d get one, three, seven, fifteen, thirty-one.
In every case, we just plug the numbers in and see what we get.
If a sequence has a domain that includes all positive integers, meaning that starting from
one you can keep plugging in numbers all the way to infinity, this is called an infinite
sequence.
All the ones we just looked at are examples of infinite sequences.
If instead, the domain stops at some integer, it will be a finite sequence.
There are also sequences that are derived not from an expression like this, but exclusively
from the previous terms in the sequence.
A famous one of these is called the Fibonacci sequence.
This one starts with a one, and then another one, and then every term after that is the
sum of the previous two.
One and one is two.
One and two is three.
Two and three is five.
Three and five is eight.
We continue in this manner to get thirteen, twenty-one, thirty-four, and so on, towards
infinity.
This sequence uses a recursion formula, meaning that we can define any term A sub N by previous terms.
In this case, A sub N will equal A sub N minus one plus A sub N minus two, with these expressions
referring to the term immediately prior the term in question, and the one two before.
Another type of sequence that does this can be found with factorial notation.
The sequence N factorial, with factorial being represented by an exclamation mark, is equal
to N, times N minus one, times N minus two, all the way until we get to one.
For example, let’s list the values of N factorial for the first few natural numbers.
We can see that in each case, we start with the number we are evaluating and multiply
it by every single smaller positive integer in descending order.
Factorials are similar to exponents, in that they only operate on the number they directly follow.
This is an important concept to understand, as we will frequently see factorials in sequences.
Say we have three over N plus one factorial.
Let’s plug in one through four to get the first four terms in this sequence.
And doing the arithmetic, these are the values we should get.
Now that we understand sequences, we can move on to sums.
Summation notation takes a sequence and then instructs you to find the sum of a certain
number of terms in that sequence.
For example, let’s just start with this first sequence we looked at, with the natural
numbers, but instead of A sub N, let’s write A sub I, since this is a slightly different
application, although technically we could use any letter here.
If we place this upper case sigma here, we can put I equals one just below it, and then
any integer above it, like five.
What this says is that we have to add up the first five terms in this sequence.
That would be one plus two plus three plus four plus five, which is fifteen.
In this case, one is the lower limit of summation, and five is the upper limit of summation.
Let’s try some examples.
How about the sum of K squared plus two to the K from one to four.
To get this, we have to evaluate the expression for each number in the domain first, and then
we add them all up.
For one, we get three.
For two, we get eight.
For three, we get seventeen, and for four, we get thirty-two.
Let’s add those up, and we get sixty.
Easy enough, right?
Now what if we go in reverse?
What if we have a sum, and we have to figure out how to express it in summation notation?
This is a little trickier because we have to recognize the pattern in the numbers, but
let’s give it a shot.
How about this one here.
One plus four plus nine plus sixteen plus twenty-five, and continuing.
What is this?
Well it’s the list of perfect squares.
We could rewrite this as one squared, two squared, and so on.
That means we could write this as the sum of N squared, from one to infinity.
We could also truncate it at any particular term, like after the fifth one, and then make
the upper limit five.
How about this one?
One half, two thirds, three fourths, four fifths, and so on.
Well the top number in the fraction is equal to N, but the denominator is always one more
than that, or N plus one.
So it’s N over N plus one.
Things can certainly get trickier.
How about three over four, six over five, nine over six, twelve over seven.
Here, the top is clearly multiples of three, so that’s three N. On the bottom, it goes
one at a time but starting with four, so that could be N plus three, where it’s always
three more than the number of the term.
As long as we always think logically in this manner, we can usually figure out even the
toughest sequences, we just look for the patterns that are present.
To wrap things up, let’s take our new understanding of sequences and factorials and show a novel
derivation for the natural base, E. Isaac Newton showed that E will be equal to one
plus one plus one over two factorial, plus one over three factorial, plus one over four
factorial, and so on to infinity.
If we work out the first few terms, we get one plus one plus one half, plus one sixth,
plus one twenty-fourth, and by now we already have E to a few decimals.
This sequence is interesting, because while it is an infinite series, it has a finite
sum, the number E. This is different from other infinite series, like the sequence of
natural numbers, which is an infinite series, and an infinite sum, because we can never
add up all these infinite numbers.
This brings up the notion of limits.
In the limit of N equals infinity, this series has a finite sum.
Sums and limits will be a big deal in calculus.
So as we can see, sequences, though they sometimes seem abstract and arbitrary, actually crop
up in nature, not just as representations of mathematical constants, but also in the
form of intricate biological designs, examples of the ways in which mathematics can produce
stunningly beautiful physical forms.
Let’s check comprehension.