Powers and the exponential function
This is the fourth article of the series. Our major task is to study the Taylor series of exp function. The series could help us understand the function from a different point of view. We will then further find a decimal estimation of e and show that e is an irrational number. Readers of this article should be familiar with the definition and properties of exp function we studied in the second article of the series.
Previously we defined exp as the inverse function of ln. In this article, we will look at this function from a new perspective. To achieve this, we need some basic knowledge about series.
A series is the ordered sum of many terms. In high school, you might already be familiar with some finite series, where there are finite number of terms being added up. Like the famous arithmetic series and geometric series:
An infinite series has infinite number of adding terms, written as
Adding to infinity is meaningful in terms of limit behaviors. We say an infinite series converges to L ∈ ℝ if as we add more and more terms in, the corresponding truncated finite partial sum could get closer and closer to L. To be more precise, we have the following definition:
The ϵ-δ definition might be a bit abstract to grasp with. But the underlying meaning of a convergent infinite series is easy. Figure 1 shows the first 50 partial sum plots for three different infinite series, their definitions are given below. We could see from their trends that only S2 is convergent, the series’ limit is 2.
Before moving on, at this stage I will further point out that a series is an ordered sum. The order in which the terms are added up matters. For a finite series, a rearrangement of the terms in the series doesn’t change its adding result. This is not guaranteed for infinite series! A rearrangement of terms in an infinite series could change the series’ limit behavior. So in general, we could not arbitrarily change the order of terms in an infinite series.
Next, we will focus on an important type of series called power series.
A power series is a sum of terms in the form of powers of x - a, it could be regarded as a function of x. For each value x takes, the series could be convergent or divergent. Notice that the series always converges at x = a, since in this case the sum is just the first term a_0. Further, it could be proved that for any power series, there is a non-negative real number R, such that the series converges in (a - R, a + R), diverges in (-∞, a - R) ∪ (a + R, +∞). R is called the radius of convergence of the power series, (a - R, a + R) is called the interval of convergence and is usually taken as the domain of the function given by the power series. For the two endpoints a - R and a + R, we need to consider the limit behavior of the given series case by case.
Here are some examples of power series and their intervals of convergence. Readers are expected to be familiar with these results. How to determine a series’ radius of convergence is out of the scope of this article.
Our main topic today, the Taylor series of a function, is a specially defined power series.
The motivation of introducing Taylor series is to use a easy-to-calculate polynomial to approximate a differentiable function f. A polynomial in the form of Taylor series (or its partial sum) turns out to be a good approximation for many functions. For example, by definition we could find out exp has its Maclaurin series in the form:
Figure 2 plots exp function and the partial sum plots of its Maclaurin series, we could see as we add up more and more terms in the series, the plot of the partial sum of the series becomes more and more close to the plot of exp in the area around 0, which means the approximation becomes better. So we are tempted to write down something like this:
For the above formula, first we need to be clear about the meaning of the equation. Given x ∈ ℝ, the left hand side of the equation is the value of exp function at x; while the right hand side of the equation should be interpreted as the limit of the power series at x. It is a point-to-point equality under the meaning of the limit of an infinite series! We then need to determine for what x the equation holds, which turns out to be a difficult problem. A function’s Taylor series is not guaranteed to be its good approximation. In general, we have the following results.
Fortunately, for our purpose, we don’t need to bother too much about the above intricacies. It could be proved that the Taylor series about a for the exponential function converges to exp(x) for all x ∈ ℝ, no matter what a is, so do the Taylor series for sin and cos functions. When a = 0, for Maclaurin series of exp, cos and sin functions, we have the following results.
Readers should be familiar with the above results, which are very important and will be used in later articles. The convergent behavior of these series is guaranteed by the Taylor’s theorem, which we will not discuss in this article.
In practice, we could only use finite partial sum of a function’s Taylor series to get an approximate value of the function at some point x. (Of course we need to first make sure the equality holds.) An important corollary of Taylor’s theorem, stated below without proof, tells us how good a partial sum approximation is.
We will demonstrate the usage of the above result by finding a decimal approximation of e with error less than 10^(-9). From our discussion so far in this series we have known that
The infinite series expression is also sometimes used as the definition of e, with functions like ln and exp defined in a different way. For our construction, it is a result of the Taylor’s theorem. If we use Nth partial sum of exp’s Maclaurin series to approximate e, we want the remainder (error) to be less than 10^(-9) (or any other precision you want), or in formula,
From Riemann integral definition and the definition of ln function, it could be proved that ln(4) > 1 = ln(e). Since ln is a strictly increasing function, this result indicates e < 4. Also remember the fact that exp is also an increasing function, then we could get exp(c) ≤ exp(1) = e < 4 for all c ∈ [0,1]. With this, to achieve the required precision, we just need
It tells us just taking the 12th partial sum of exp’s Maclaurin series is enough to give us an approximation of e with error less than 10^(-9), the result approximation is
Though we are now able to get an approximation of e with any precision, any approximation itself is a rational number, so it doesn’t help in revealing the irrationality of e. In the last part of this article, we give the proof that e is an irrational number.
We have gone through a long way in this article. The key content of it is the Taylor series of exp function, with the help of which we could prove many important properties of e and understand exp in a different and useful perspective. In the remaining three articles of this series, we will enter the field of complex numbers. Powers and the exponential function will remain to be our main topics, their properties could be expressed in a more unified way in the complex field.
