has puzzled mathematicians for ages. Already 4000 years ago, both Egyptian and Babylonian cultures had figured out that the ratio of the circumference to the diameter of any circle is a constant.
On the other hand, when it came to determining this constant, both societies were quite far off with their estimations. The task may sound simple, but the answer is not that obvious.
The reason is that is an irrational number, and no number of decimals is ever enough to represent it exactly. That being the case, it has been of utmost interest to approximate its value for practical use.
Until quite recently, the most accurate method used outer and inner polygons of increasing numbers of sides to bound the area of a circle. The areas of such polygons can be calculated exactly, from which an interval for is derived.
If we let the number of sides grow larger and larger, we form an infinite series of approximations that just gets more and more accurate. Using this method with a polygon of sides, is found exactly up to the 37th decimal place.
If you add 5 natural numbers in a row, you will have a sum. Lets call this sum
If you add infinitely many natural numbers, then you would have a series
A series is nothing but an infinite sum
A series can either diverge or converge. If a series diverges, it means that it will be equal to either or . If the series is convergent, it will become closer and closer to a certain specific number. It might not reach it but it will become arbitrarily close. For example say we have the following series:
This series will never reach 1. But it will become arbitrarily close. We say that this series is convergent equal to 1. As for the case above we even go as far as to write:
for most application. Example of divergent sum is:
This sum is divergent and we sometimes write:
to show that it is divergent. In the following text we will go through some test to check if a series is convergent or divergent. We will see that:
and is therefore divergent, we will also see that
and is therefore convergent. Finding if the two cases above is divergent or convergent will be done using many tests. The simplest of these is the test. For the series below, if , the series converges. If , the series diverges.
Sequences and their characteristics
A sequence is an ordered list of elements which has a beginning but no end. The most elementary of sequences is the sequence of positive integers . We call the elements of a sequence terms. The index usually starts on or on .
There are three ways of presenting a sequence:
we can give a list followed by ..., if the terms follow a nice pattern,
we can give a formula for finding from the previous terms,
we can provide a formula for as a function of .
When it comes to notation, we'll denote a sequence as .
A very famous sequence is the Fibonacci numbers, defined as:
This sequence is documented as early as years ago, and can be found in nature all over, for example in the shapes of chicken eggs, chameleon tails and romanesque broccoli.
A sequence is said to be increasing if for all we have , and upper bounded if there is a number that is at least as big as the biggest .
Likewise, a sequence is said to be decreasing if for all , , and lower bounded if there is a number that is at least as small as the biggest .
A sequence that is both upper and lower bounded is just bounded. It stays obediently between some and some .
Convergence of sequences
If as we let go to infinity, the sequence goes towards some number , we say that it is convergent, otherwise it is divergent.
Some divergent sequences goes to infinity as grows, some just jumps around, not approaching any number at all. One example of the second case is:
This is an alternating sequence, meaning that two consecutive terms always have opposite signs. It jumps forever between and .
All sequences which are both upper bounded and increasing, or lower bounded and decreasing, converge. This is quite intuitive: a sequence forever growing but not going over some number needs to get infinitely close to as the sequence gets longer and longer.
Some more examples
The above sequence diverges towards , while the following one diverges towards :
The sequence above is increasing and upper bounded, so it converges:
Finally, this sequence is decreasing and lower bounded, converging towards :
Series and convergence
A sequence to start with
Here is a sequence:
Notice that the next term is the last one doubled. Thus, calling the th term gives
Further, as this goes for all the terms, we can write:
Note that in this case is . Go ahead and test the formula's correctness for the first couple of terms!
Series and partial sums
Let be a sequence, and let be the sequence where each element is defined as:
Then, we define the series as:
That should look confusing, at first. But all we say is that the series is what we get, if we sum all the elements in the sequence . So, the series is actually a sum.
The numbers are the partial sums of the series:
The partial sums can be written more compactly as:
A small refresher: the big Greek zigzaggy letter is the symbol for a sum. The at the bottom together with the on the top means: we take the sum of the terms with index to of what's written after.
In the example we had, the partial sums would be:
A partial sum of a series is the sum of the first terms of the sequence
Convergence and divergence of series
So, a series is a sum of an infinitely long sequence. But how can that ever be a number? If we put infinitely many elements together, don't they always sum up to infinity?
Well, it turns out that that's not the case. Some dark magic feature inherent to mathematics make that some sequences, though infinite, sum up to a number.
These are series where the limit:
exists. We call them convergent. Series where this limit shoots off to infinity are called divergent.
What this means is that if the series is convergent, we can find an upper bound so that the sum always stays below it, as we add more and more terms of the sequence .
Divergent series, on the other hand, will always step over any bound, even if we try real hard to be generous.
Note that even if the sequence converges, it does not mean that the series does. In fact, we need the sequence to converge towards zero for the series to converge, and not even this is enough.
In the coming notes, we'll go through this and two other methods for determining if a series converges.
Geometric series are series of the form:
The terms constitutes a geometric sequence, so that for some .
These series are in a group of just a few types of series which we can calculate, as long as .
The -series for paper size
The -series is a standardised system for paper sizes, where the biggest one is called . In , there fits papers of size . In , there fits , or , or and so on.
This means we can write:
The sum to the right is a geometric series, and it's equal to . We will show this in a moment.
So, if we put all the papers of size together, we get:
Neat, isn't it?
If this didn't convince you, just wait for the surprisingly visual proof below.
Why is the sum equal to ?
For a general geometric series:
the is the ratio of the series. In the paper case, .
We'll show why. We start with performing a trick, which will lead us to the sum of the series. Observe these two sums:
The first one is a partial sum of the series.
Subtracting , observe that most terms just eliminate each other. Finally, we get:
Shuffling some things around, we get:
This is the formula for the partial sum of a geometric series with terms.
As we let go to infinity, the disappears (we required ). Thus, when , it gives the sum of the series:
So, using and in the above formula, we see that the sum of all the papers smaller than is:
So we can fit an infinite sum into one sheet of paper!
Calculate the geometric series below:
Note that the formula starts at . Thus, we can not use the formula for geometric sum straight away. But using the following trick, we can transform the expressions into two sums:
We can now use the two formulas for sums and partial sums:
The term test
The compound interest you've earned since depositing an amount into your savings account is calculated by the formula:
where is the interest rate given as a percentage, and how many times the bank computes interest, which is usually once per day.
Is there a limit to how much money we can make from a given amount?
We can rephrase the question as: If , does the series converge?
The answer is no, and we can prove it using certain techniques known as convergence tests.
The term test
Due to its simplicity, the first test we usually employ to examine the convergence of a series is the term test.
If the terms in the series do not approach zero, the series will diverge
The term test
Let be the th term in a sequence. Now if
To see why this is true, imagine what would happen if the limit was not zero, but some other value . Then, when , , and the sum of the terms will approach , where is the number of terms we consider.
Since there is no limit to how large can be, the sum will become infinitely large or small, depending on the sign of . Consequently, the series diverges to .
Note that the theorem states an implication and not an equivalence. In other words, just because this limit happens to be zero we cannot conclude that the series will converge.
Let's return to the formula for calculating compound interest:
Letting , we get the series:
Here, is a constant and a small percentage, and so . Therefore, , the th term of the sum, will tend to 0 with growing .
As a consequence, the term test is inconclusive, and we need to go on to additional testing. We will study additional convergence tests in the coming lecture notes.
diverges since if we apply the term test, we see that:
If we apply the term test to the series:
then we see that:
Since the term test gives us the limit 0, we can't draw any conclusion about divergence or convergence, but have to employ other tests. This series is called the harmonic series and is in fact divergent.
Things are starting to get quite abstract now, so let's just take a step back and recap where we currently are.
A series was just this sum with an infinite number of terms:
A word of warning: we're not writing , because there isn't a last term . So we literally want the "" part.
In calculus, we often want to know whether a series has a particular value. Maybe a series has a value of or . Series can also shoot off to infinity, or negative infinity!
Series pop up all the time in the real world. There are numerous applications in economics, physics - and even to gambling. For example, the idea of a series can be used to determine whether a bet is likely to pay off or not.
The ratio test
The ratio test allows us to infer whether a series has a particular value, as opposed to or . In most real-world applications, we're dealing with the first kind of series.
The ratio test compares two consecutive terms, and
Ok so, you've got a series. Now have a look at the ratio of two consecutive terms:
If the limit is , then the series is convergent. Each term gets smaller and smaller, so the series will grow slower and slower. Ultimately, it'll converge towards some value.
But if the limit is , each term gets bigger and bigger. The series will explode - growing quicker and quicker. This means that:
tends towards !
But what if the limit is ? We can't tell. As we've seen before:
is , whereas:
assumes some value. So the ratio test isn't like a theory of everything, and you need to use it with care.
Using the ratio test, we can see whether the series:
will converge. Firstly, calculate the limit:
Thus, the series will converge according to the ratio test.
Use the ratio test to if the series below diverge or converge.
The ratio test is performed as follows:
Next we multiply both numerator and denominator with k:
After this we divide both numerator and denominator with k:
The limit becomes 1 and therefore we cannot conclude if the series converge of diverge.
The p-test for series
A -series is one where the terms consist of divided by the index raised to some power :
One special case of a -series is when , which is known as the harmonic series and looks as follows:
The harmonic series is a special case of a -series, and it diverges to
There is a very simple technique we can use to determine whether a -series converges or diverges, called the -test.
converges if , otherwise diverges to .
Note that the harmonic series, for which , therefore diverges to .
To determine whether the series:
diverges or converges, we first need to compare the summand to:
This is true since if we make the denominator smaller, the fractions gets bigger. Thus, we get:
According to the -test, then:
converges. Since this series is always greater than or equal to our series, our series cannot diverge, so it must converge.
diverges since, we compare our summand to:
Thus we have the following:
Since the series:
is always less than or equal to our series, our series must diverge since:
diverges according to the -test.