Revisiting Convergence

We are in the middle of the second pit stop on our journey of exploring complex numbers. We have reached a point where I can introduce us to some infinite series, as mentioned in the previous post. Of course, we have encountered infinite series when we studied e as well as when we studied π. An infinite series is simply the summation of all the terms of an infinite sequence.

In case the preceding sentence was confusing, let us recap a bit. A sequence is a succession of numbers generated according to a well defined rule. A sequence can have a finite number of terms or it could be an infinite sequence. A series is formed by adding the terms of a sequence. An infinite series can converge, diverge, or oscillate. A classic case of such a series is the geometric series. We considered the geometric series in Naturally Bounded?, the second post in the series on e. There, I only considered a single case where the common ratio r was 1/2. That series converged. In general, a geometric series will converge if the common ratio lies between -1 and +1. In all other cases, it will diverge. And if r is negative, the series will also oscillate, converging while oscillating if r is between -1 and 0 and diverging while oscillating if r is less than or equal to -1.

Conditions for Convergence

What this means is that, when we are faced with an infinite series, there will be conditions under which the series will converge. When these conditions are not met, the series will diverge, either oscillating or not oscillating.

So suppose we have an infinite series

We can define the partial sum of n +1 terms as

Now, if

we can say that the series converges and that it converges to the finite sum of S_∞. In general, a necessary but not sufficient condition for a series to converge is that each term approaches zero. That is,

However, this test is not sufficient to guarantee that the series converges. For that, we have to apply many other tests. However, this is not what I wish to focus on. For now, let us simply assume that the series we will introduce are convergent. Will will then be able to obtain some rather intriguing results.

Infinite Series for e^x

So, suppose that

is a convergent series. Now, from the previous post we know that the derivative of e^x is e^x. From the post that preceded it, we know that the derivative of xⁿ is nx^{n – 1}. So, if we differentiate the above equation, we will get

We can differentiate this again to get

We can continue in this manner ad infinitum. However, let us see what the value of all of these are when x = 0. When x = 0, e^x = e⁰ = 1. Also, all terms with x^k will become 0, leaving us with

Rearranging each equation, we get

Recognizing that n! = n(n – 1)(n – 2)···3·2·1, and defining 0! = 1, we can write the above as

From this we can write the infinite series for e^x as

In the post Infinitely Expressed we looked at the speed with which a series converged. How does this series compare? The table below gives us the relevant information

What we can see is that, as n increases, the error decreases by about one order of magnitude for each increased value of n. In the aforementioned post, we had seen that this series does not converge very quickly and we introduced other series that converge much more rapidly. However, here we are not concerned with the speed with which the series converges. Rather, the simplicity of the terms themselves provide a strange allure. From the image at the start of this post we can see how each additional term makes the resulting series more closely approximate the graph of y = e^x.

Infinite Series for sinx

Suppose now we attempt to obtain a similar infinite series for sinx. So let us consider that

When we repeatedly differentiate the equation we will obtain the following

Now, if we put x = 0, we will have sinx = 0 and cosx = 1. Then we will obtain

Rearranging these equations we get

From this we can obtain

Infinite Series for cosx

In a similar way, for the cosx function, we can obtain

Setting the Stage

So far we have obtained infinite series expressions for e^x, sinx, and cosx. This gets us to the point where we can use these series to obtain the exponential form of the complex numbers. We will deal with that in the next post. Hence, this pit stop too has come to an end. We have, quite obviously, only dealt with those concepts in calculus that directly play a role in determining the exponential form of the complex numbers. Once I have concluded with the series of posts on complex numbers, we will revisit calculus. But till then I hope that this short pit stop has been helpful in your reaching the realization that calculus is not something to fear.

Rewinding a Bit

We began our exploration of calculus last week with The Sky is the Limit. This is the second pit stop in our exploration of complex numbers. The first pit stop dealt with trigonometry. In the previous post we had looked at the idea of the limit, which I said was a central concept in the study of calculus. In this post I wish to obtain a few standard results that we will be able to use in our exploration of complex numbers.

In the previous post, we saw that the limiting value of the gradient of the secant through a fixed point on the graph of a function as another point approaches the fixed point is the gradient of the tangent at the fixed point. In the context of calculus, this is known as the derivative of the function at the point.

Definition of the Derivative

Suppose we have a function y = f(x). Then the derivative is denoted by

Per the limit definition of the derivative, we can say that

Note that the quantity inside the limit is simply the gradient of the secant at the point (x,f(x)). As h approaches 0 the gradient of the secant approaches the gradient of the tangent (i.e. the derivative) and the limiting value of the gradient of the secant is the gradient of the tangent.

Algebra of Differentiation

In what follows, u and v represent function of x.

The Sum Rule

The derivative of the sum of two functions is equal to the sum of the derivatives of the two functions. Symbolically, we state (u + v)’ = u‘ + v‘. This can be obtained as follows

Difference Rule

The derivative of the difference of two functions is equal to the difference of the derivatives of the two functions. Symbolically, we state (u – v)’ = u‘ – v‘. This can be obtained as follows

The Product Rule

The derivative of the product of two functions is equal to the sum of the product of the first function and the derivative of the second and the second function and the derivative of the first. Symbolically, we state (uv)’ = uv‘ + vu‘. This can be obtained as follows

The Quotient Rule

The derivative of a quotient is equal to the difference between the products of the denominator and the derivative of the numerator and the numerator and the derivative of the denominator divided by the square of the denominator. Symbolically, we state (u ÷ v)’ = (vu‘ – uv’) ÷ v². This can be obtained as follows

Derivative of xⁿ

Having obtained the results for the algebra of differentiation, let us derive our first derivative. Let us consider the function y = xⁿ. Per the definition of the derivative we have

While I have only derived the result for positive integral values of n, the result can easily be obtained for all rational values of n. But what does this mean? Let’s take some concrete cases.

Suppose f(x) = x³. Here we can see that n = 3. Hence the derivative will be given by f'(x) = 3x². This means that, at the point where x = 2, the gradient of the tangent is f'(2) = 3(2)² = 12. Similarly, for the function f(x) = x^2/3, n = 2/3. Hence, the derivative will be given by f'(x) = (2/3)x^-1/3.

Derivative of sin x

Now, we have seen that the polar form of complex numbers involves the trigonometric ratios of the argument of the complex numbers. This means that it is quite likely that the derivatives of the sine and cosine functions will be needed in our further exploration of complex numbers. So let us obtain the derivative of the sine function. Suppose y = sin x.

Limit of sin x ÷ x as x → 0

Let us first obtain the limit of sin x ÷ x as x approaches 0.

The Sandwich Theorem

Here we need to invoke a theorem known as the Sandwich theorem. Here it is

Let us return to our derivation of the limit and use the Sandwich theorem.

We now have enough under our belts to obtain the derivative of sin x. Let’s proceed.

In a similar way we can show that (cos x)’ = -sin x.

Next Steps

In the derivations above, I have not shown all the ‘steps’ but have skipped some that are, in my view, decipherable by students who are willing to put in some additional effort. If any of the steps above is unclear, please do reach out to me. At the same time, I realize that, while the above results are old hat for students who have studied calculus, in many cases, these students have not been shown how the results are obtained. Shockingly, I have had students come to me in the twelfth grade having been introduced to differentiation but having no idea how the results are obtained. Due to this, I think this post is already quite heavy. So I will draw it to a close. In the next post, we will consider the derivatives of the exponential and logarithmic functions. Till then, be responsible and don’t drink and derive!

The Rationale

We are in the middle of a series on complex numbers. In the previous post, Anticipating the Exponential Whirligig, we had come to a critical point, where I claimed that there was another way of representing complex numbers that is even more powerful than the polar form, which was itself more powerful than the Cartesian form. However, this form is derived using calculus. I could have simply thrown the form in your faces, hoping that you would take me at my word. And while you may be willing (I hope) to take me at my word, this would not have given you any insight into why the form exists nor why it works the way it does.

In fact, at the end of the previous post, we had actually reached a somewhat counterintuitive idea, namely that, since the modulus (r) gives the ‘size’ of the complex number, the argument (θ), which represents the rotation, must be what is expressed in an exponential form. Since this is counterintuitive, it would not do for me to simply state a result and proceed. I need to show how the result is obtained. Hence, we are forced to take another pit stop, this one dealing with calculus.

What is Calculus?

Of course, if we ask, “What is calculus?”, different sources give us different definitions. For example, Britannica states, “calculus, branch of mathematics concerned with the calculation of instantaneous rates of change (differential calculus) and the summation of infinitely many small factors to determine some whole (integral calculus)” (Italics mine.) Wikipedia gives us, “Calculus is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithmetic operations.” (Italics mine.) The MIT course Calculus for Beginners states, “Calculus is the study of how things change. It provides a framework for modeling systems in which there is change, and a way to deduce the predictions of such models.” (Italics mine.) The MIT definition could be said to be only the first sentence. However, this definition does not allow us to recognize that there are two facets to calculus, something that I believe is essential to know right from the start.

While these definitions are not wrong per se, I think they somewhat miss the point (Wikipedia), beat around the bush (Britannica), or give extraneous information that is confusing (MIT). For example, unless I know what it means to says that “geometry is the study of shape” or that “algebra the study of generalizations of arithmetic operations,” I will not have a cluse about how calculus is the study of continuous change! Similarly, if I have no idea why we would need to calculate instantaneous rates of change (or indeed what that means) or sum infinitely small factors (or indeed what that means), I have no idea what calculus deals with. Finally, why the provision of “a framework for modeling systems in which there is change” is included in the definition of what calculus is when it is something that calculus does just boggles me.

I would give the following definition: “Calculus is the study of the rate and the development of change.” I think my definition is superior because it is sleek, using just 9 words as opposed to 18 (Britannica), 25 (Wikipedia), or MIT (27). Here I have counted only the predicate clauses given in italics since that is what forms the definition. Of course, the study of rates of change is known as differential calculus, while the study of the development of change is called integral calculus. In this pit stop, I will be dealing with differential calculus because that is what is immediately relevant to our understanding of complex numbers. I will deal with integral calculus as a later date. So let’s get started.

Initial Intuition

Suppose we have a straight horizontal road. A car on the road moves from traveling at 10 m/s to 30 m/s in 5 seconds. The previous statement only gives us the starting and ending states of the car on its 5 second ‘journey’. We know nothing concerning what it was doing (i.e. its speed/velocity) or where it was (i.e. its position/displacement) in the intervening period between times t = 0 seconds and t = 5 seconds. For example, the figure below shows four of the infinitely many possible variations of v (velocity) and t (time) that satisfy the conditions given earlier.

While all four journey profiles have the same initial conditions (t = 0 s, v = 10 m/s) and final conditions (t = 5 s, v = 30 m/s) all of them differ at every other point from every other profile. Quite obviously, it will not suffice to simply think of initial and final conditions, not if we want to be able to give a thorough description of the profiles. For example, we could draw the tangents at t = 2 s to get

Of course, the ‘tangent’ to the first graph, which is linear, is the line itself. Hence, only three other distinct tangents are visible. What we can easily observe is that the gradient or slope of each line is distinct from the others. Since we have plotted speed/velocity on the vertical axis and time on the horizontal axis, the gradient would indicate the rate at which the velocity is changing with respect to time when t = 2 s. Of course, the question is, “How do we obtain these gradients?” This is where calculus comes in.

Introducing the Limit

To understand calculus, whether differential or integral, we need to grasp the concept of a ‘limit’. Of course, here we are dealing with the limit in the context of differential calculus. To get a feel for what this is before we begin a more formal treatment, click on the link in the caption on the figure below. This will take you to a Geogebra app. It contains the plot of the graph of a function in red. There is a tangent to this graph at the point A in blue. Another point, B, lies on the graph and the secant connecting A and B is drawn in green. Click and drag on the point B. This will allow you to move B along the graph. As you do this, the orientation of the secant (green line) will change. It will, in other words, rotate about the fixed point A.

Now, since the point A is common to both lines, the only difference between them is their orientations or gradients. You will observe that, as you move B closer to A, the secant will rotate about A with its orientation becoming closer to the orientation of the tangent. In fact, you can compare the gradients in the app itself. In the figure above, the gradient of the tangent is 15.33541, while that of the secant is 8.27131. You will note that there is a quantity h below the two gradients. In the figure above h = 1. This quantity represents the horizontal distance between points A and B. In other words, it is the difference in the x coordinates of the two points.

The Gradient of the Secant

Now, in the figure above, x is listed as the x coordinate of A. This means that the x coordinate of B will be x + h. Now, since the graph is that of the curve y = f(x), the corresponding y coordinates will be f(x) and f(x + h).

Now the gradient of the line joining the points (x₁,y₁) and (x₂,y₂) is given by

Hence, the gradient of the secant through A and B will be given by

Of course, this only tells us what the gradient of the secant is. We are not yet at the point of determining the gradient of the tangent, which, if you recall, from the earlier exploration of the 5 second journey, will give the rate at which the velocity is changing with respect to time at any given instant.

At this stage, we have no idea what the function is. Hence, things are quite abstract. In order to allow us to attach some numerical quantities to the above ‘formula’, let us consider the function f(x) = x². I have intentionally chosen an easy function so we can get to the meat of the matter quickly. Let us also consider the x coordinate of A to be 2. Hence, the gradient of the secant will be given by

We can either expand the parentheses in the numerator or factorize the numerator to simplify as shown below

In the step above, the line in red is the approach with expansion, while the one in green is the approach with factorization. As the color scheme indicates, I prefer the factorization approach even though both give the same results.

Revisiting the Limit

What we have obtained so far is that, for the graph of y = x², the gradient of the secant joining points A and B with x coordinates 2 and 2 + h respectively is 4 + h. Now, we saw that, when B approaches A along the curve, the secant rotates about point A with its orientation becoming closer to the orientation of the tangent. So let us see how the gradient of the secant varies as h varies. The table below shows the variation on both sides of x = 2.

Note that there is no secant defined for x + h = 2, because that corresponds to h = 0, which would make the expression for the gradient indeterminate since the result would be 0 ÷ 0.

Introducing Indeterminacy

Here, a note on indeterminacy is in order since I have had teachers and colleagues who have used the word ‘undefined’ in cases when they should have used ‘indeterminate’. The word ‘undefined’ is normally reserved for cases involving division by zero since it is meaningless to divine by zero. However, this is only when we are attempting to divine a finite, non-zero quantity by zero. When we attempt to divide zero by zero, we face a quandary.

You see, division of a non-zero quantity by zero is, as we have just seen, undefined. However, division of zero by a non-zero quantity results in zero. So in the expression 0 ÷ 0, which of the zeroes takes precedence? Since the whole expression for a function that yielded the problematic 0 ÷ 0 is important for the definition of the function it is impossible to determine which zero should take precedence. Hence, the expression is indeterminate.

Similar indeterminate forms yield expressions like 0 × ∞, ∞ – ∞, ∞ ÷ ∞, 0⁰, ∞⁰, and 1^∞. In each of these, there are two parts and it is impossible to determine which part should be given precedence. Hence, these forms are indeterminate.

The Meaning of the Limit

Returning to the table, we can see that, as x + h gets closer and closer to 2, the gradient, m, gets closer and closer to 4. So we say that the limit of the gradient as h approaches 0 is 4. It is crucial to observe that this statement does not mean. It does not mean that the gradient of the tangent can be determine by putting h = 0 in the original function. Neither does it mean that the gradient of the secant becomes the gradient of the tangent when B coincides with A.

What the statement about the limit declares is something about the behavior of some parameter, in this case the gradient of the secant, around a point where it is technically indeterminate. It says that, by making h arbitrarily close to 0, we can obtain a secant whose gradient is arbitrarily close to the gradient of the tangent. And since the gradient of the secant approaches 4, it must mean that the gradient of the tangent is 4.

A Glimpse Ahead

The limiting process is the cornerstone of both branches of calculus. We have actually already seen it in this blog in a more informal manner. In A Piece of the Pi, which was the first post in the series on π, we had seen that early attempts at approximating the value of π involved regular polygons with increasing number of sides. This was nothing but a limiting process since the area of the circle was always squeezed between the area of the inscribed polygon and that of the circumscribed polygon. As the number of sides increased, the areas of the two polygons approached each other and in the limiting case would have become the area of the circle.

Today, we have seen a slightly more formal look at the limiting process. Of course, we only considered one special function, f(x) = x², which was relatively easy to analyze. However, we need to be able to use the limiting process for functions of all sorts – algebraic, trigonometric, inverse trigonometric, exponential, and logarithmic. And we know that we are aiming for something in the exponential side of things for the exponential form of complex numbers. In the next post, therefore, we will continue our journey into calculus with a look at some algebraic and trigonometric results. Till then, the sky’s the limit.