Sequences are simply lists of numbers, that’s it. The only rule is that we keep the different items in the list in order. For example, the sequence is not the same as because the and have switched places.
The idea behind the definition for a limit is an intuitive one, although this is somewhat hidden by the terminology. Suppose we have a long list of numbers like . The numbers never actually reach zero, since one over something is never zero. However, we can see at the same time that the numbers get closer and closer to zero. So the limit of the sequence is zero, even though the sequence never quite makes it there. The definition is meant to recognize precisely this kind of situation.
The points keep getting lower and lower and no matter how small a number you can think of, they will eventually get smaller than that number. So the limit is . In math terms, for any , there is a value so that each term is no bigger than .
In more advanced analysis, mathematicians consider sequences in a slightly more general context. Think about two-dimensional space. This is the set of real number pairs, like or . Three-dimensional space is the set of real number triples, like or . Similarly, dimensional space is the set of real number tuples like . Sequences then are like dimensional space, the set of tuples like: . This kind of formulation allows for analysis of sequences using more topological or geometric terms like compactness and boundedness.
Any kind of sum, like can be written more compactly as:
When the number above the big is (instead of as it is above), we call this sum an infinite series. Adding together an infinite number of terms usually leads to disaster. But occasionally, if the terms get small fast enough or if enough of them are negative, then the sum can be something very interesting.
One classic example of a series is Zeno’s Paradox which asks whether an arrow will ever reach its target given that it only ever travels half the remaining distance. This, of course, boils down to summing a geometric series with and can be viewed geometrically as follows: Consider a square whose total area is . Then the first term in the series is just half of this square, and the next term is just half of the remaining area. And the next term is half of the remaining area and on and on so that it becomes clear we are simply filling the entire square. This a geometric demonstration of the fact that , a result that is also obtained by using the formula for the sum of a geometric series.
Series Without Negative Terms
The unique quality that makes non-negative series so attractive is that they have clear geometric interpretations. When a series is never negative, we can graph the points. Then the series is just a kind of course Riemann Sum where the rectangles all have width equal to .
As a very telling example, consider the harmonic series:
We can visualize each term as the area of the rectangle that has . To see this, we zoom in on the first points look at the corresponding rectangles along with the function corresponding to the lower sum and the function corresponding to the upper sum:
This plot shows why it is clear that the sum must diverge. This is because the sum is the area of the pink rectangles above, and this area is between the areas of and . However, both of these integrals are infinite since and .
We can also see geometrically why the integral test works, since the series is simply a course Riemann Sum for the integral. If the integral with the same form and limits diverges, then so too must the sum since as we move farther and farther to the right the upper and lower sums are both indistinguishable from the integral. That means that the difference between the series and the integral must be only the finite piece on the left where the function is significantly different from its Riemann Sum, but any finite number added or subtracted cannot affect convergence/divergence.
The Simplified Comparison Test has a similarly simple geometric interpretation. If we are interested in whether a series converges or diverges. We might look for a simpler series that “is asymptotically similarly to ”:
This is the same as saying that the areas of corresponding rectangles for and as we move farther and farther out approach the fixed ratio . But then if the area corresponding to the sum of the is finite or divergent, then so too must the area for the sum of the be finite or divergent respectively.
We’ve seen that The Harmonic Series is divergent, however it is interesting to note that the sum gets big excruciatingly slow. For example, the first hundred terms only add up to about and the first thousand only add up to about . In fact, it takes over terms for the sum to surpass . Another interesting question to ask is whether the series will diverge when we take out certain terms. For example, suppose we remove any terms that have a in the denominator. This is not very many terms since it is only in the first and in the first and so on. Because we are taking infinity and removing so little, it seems like we should still have infinity. However, it turns out that removing these is enough to cause the series to converge!
Series With Odd or Even Negative Terms
Alternating Series are everywhere in math and science. So it is extremely important to understand how to manipulate them. For example, the functions and are all alternating series in the variable .
What about the simple alternating series What is this sum equal to? Well, you may see that after one term the sum is one whereas after two terms the sum is zero. Then the sum is one again after three terms and zero again after four and so on. So the partial sums fluctuate back and forth between and . This is not a convergent series since the sequence of partial sums is , which has no limit.
On the other hand, you may have always suspected that this series sums up to . I know that I have. Here’s a proof that seems to vindicate this suspicion. (Can you find the error?)
- Solve the algebraic equation:
- You should find that
- Now use iteration to solve this equation in a different way. This is a technique that is very important in science. Take the equation: and plug in for , since they are equal after all.
- You should get the equation
- Now repeat this for the new on the right hand side:
- You should now get the equation
- Repeat once more for the new on the right hand side:
- You should now get the equation
- Repeating indefinitely we obtain that
- But so if we get rid of the parenthesis we’ve shown that:
The error in this proof was in using the series This is a divergent series and by grouping the terms appropriately we can make it be anything we like.
People have claimed throughout history to have found very interesting results in math by using divergent series. However, these are often the result of playing tricks that cannot be generalized to the larger arena of mathematics. In fact, one of math’s most incredible genius’ Niels Henrik Abel was moved by such claims in to write that, “Divergent series are the invention of the devil, and it is shameful to base on them any demonstration whatsoever.”
Ratio Test, Root Test, and Summary of Tests
The only path to mastering when series converge and diverge is experience. Furthermore, once you start to understand which series converge and which diverge you will be able to understand the behavior of functions that are extremely important throughout science. When the function looks like a series you are familiar with, you know what to expect and this will provide indispensible intuition.
One example comes from particle physics and the area of quantum field theory. Certain problems arise when one looks in detail at problems of particle interaction, since the summations or integrals tend to diverge. However, when this divergence is similar to that of the harmonic series , then this can be canceled off later through a process called renormalization. The ability to quickly tell when the divergence is like this (as in quantum electrodynamics) and when it is not (as in gravitation) is cherished in the physics world.
It is also somewhat relieving, and interesting, to note that convergence and divergence can essentially always be boiled to down to comparison with a test series or a geometric series. This is at the heart of all the tests described in the chapter. The hard part is just figuring out how to simplify the terms so that they look like something familiar.
Consider a vector in two-dimensional space like or a vector in three-dimensional space like . The vectors and are called the basis vectors for the space. This means that any vector can be written out as a sum of these with some coefficients. Now, the letters , and are not a very good choice if we want to go into higher dimensions. Instead, in n-dimensional space we write that:
where the are numbers and the basis vectors are the . This can be written more compactly as:
We can even imagine an infinite-dimensional space where there are infinitely many basis vectors and the general vector looks like:
Now imagine that the vectors are not little arrows but are more abstract. In fact, think of the vectors as being functions like or . Any function can be written out like a vector:
where the are numbers (the components of the vector) and the are the basis vectors or basis functions! That is to say, the function may be anything at all, but the right hand side is pretty simple. It consists of regular old numbers and some set of simple functions . Do you recognize some simple functions that we might choose for the ?
This choice gives us the power series around zero (aka the Maclaurin Series) for the function . Then the coefficients of expansion are just:
Now, one important property of a space like dimensional space or the infinite-dimensional function space is the ability to measure distance between points. In dimensional space we use a kind of iterated Pythagorean Theorem to give that the distance between the tips of the vectors and is just:
In the infinite-dimensional function space it is customary to measure the distance between the functions and by the following kind of continuous extension of the Pythagorean Theorem:
where the limits of integration and will depend upon what kind of functions we have. Notice however that this definition is not in terms of the coefficients or .
In terms of this distance measuring business, a good basis for the space will have a few very important properties: The basis vectors should all have unit length and they should be directed perpendicularly. The first condition is referred to generally as the normalization condition and the second is called the orthogonality condition, so that a basis satisfying both of these is called orthonormal. The basis for dimensional space, for example, is orthonormal since the vectors are all perpendicular to one-another and each has a length of one.
However, it turns out that the power-series basis for functional space is neither orthogonal nor normalized in terms of the standard distance formula given. However, we can build an orthonormal basis from the basis using a canonical process called Gram-Schmitt Orthonormalization, and the result is a set of polynomial basis functions called the Legendre Polynomials.
Another choice of basis for functional space is the trigonometric functions of varying frequency or giving a function the expansion:
This is called the Fourier basis and the expansion is called a Fourier Series for . It turns out that this is better basis than the power series basis since it is already orthogonal and all that is needed is a little number in front of each basis function to make them unit length.