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Section 2.1 Power Series

Subsection 2.1.1 Series as Functions

Our two main examples for comparison in the previous chapter were the geometric series and the \(\zeta\) series. Both converged for certain values of the parameter; \(|r|\lt 1\) for the geometric series and \(p >\) for the \(\zeta\) series. To start this section, I'd like to re-interpret these two series. Instead of thinking of a whole family of different series which depend on a parameter (\(r\) or \(p\)), we can think of each family of series as a function of the parameter. In this view, we have only one series, but the series produces a function instead of just a number.

For the geometric series, this new perspective defines a function \(f(x)\text{.}\)

\begin{equation*} f(x) = \sum_{n=0}^\infty x^n \end{equation*}

The domain of this function is \(|x|\lt 1\text{,}\) since those are the values of the parameter (now the variable) where the geometric series converges. We even know what this function is.

\begin{align*} \amp f(x) = \sum_{n=0}^\infty x^n = \frac{1}{1-x} \amp \amp |x| .\lt 1 \end{align*}

In this way, the geometric series now defines the function \(f(x) = \frac{1}{1-x}\text{.}\) The domain restriction of the function is determined by the convergence of the series: a point \(x\) is in the domain of the function if the series converges for that choice of \(x\text{.}\)

We can do the same with the \(\zeta\) series. The reason we called them \(\zeta\) series is that the associated function is called the Riemann \(\zeta\)-function.

\begin{equation*} \zeta(p) = \sum_{n=0}^\infty \frac{1}{n^p} \end{equation*}

The domain of this function is \(p \in (1,\infty)\text{,}\) since that is where the series converges. (In other branches of mathematics, the domain of \(\zeta\) is extended in new and interesting ways. The vanishing of the \(\zeta\) function is the subject of the famous Riemann Hypothesis, an important unsolved problem in modern mathematics.)

In general, an infinite series can represent a function \(f(x)\) when the terms \(a_n\) of the series also depend on \(x\text{.}\)

\begin{equation*} f(x) = \sum_{n=1}^\infty a_n(x) \end{equation*}

Notice that the variable of the function, \(x\text{,}\) is not the index of the sum \(n\text{.}\) These two numbers are different and must not be confused. The domain of this function is the set of values of \(x\) for which the series converges. Instead of previous domain restrictions, involving division by zero, square roots and other problems, domain restrictions are now all about convergence. For these series, convergence is no longer a yes/no question. Instead, it is a domain question: for which real numbers does the series converge?

Subsection 2.1.2 Definition

A polynomial \(p(x)\) of degree \(d\) can be written as a finite sum in sigma notation.

\begin{equation*} p(x) = \sum_{n=0}^d c_n x^n \end{equation*}

The terms involve powers of the variables (\(x^n\)) and coefficients of those powers (\(c_n\)). What if we let the degree become arbitrarily large, going to infinity?

Definition 2.1.1.

A series of the form

\begin{equation*} f(x) = \sum_{n=0}^\infty c_n x^n \end{equation*}

is called a power series. The real numbers \(c_n\) are called the coefficients of the power series. The whole expression \(c_nx^n\) is still the term of the power series.

The full definition is slightly more general. The previous series was a power series centered at 0. We can centre a power series at any \(\alpha \in \RR\text{.}\)

Definition 2.1.2.

A series of the form

\begin{equation*} f(x) = \sum_{n=0}^\infty c_n (x-\alpha)^n \end{equation*}

is called a power series centered at \(\alpha\). The numbers \(c_n\) are still called the coefficients and the number \(\alpha\) is called the centre point. The whole expression \(c_n(x-\alpha)^n\) is still the term.

Subsection 2.1.3 Radii of Convergence

Polynomials were defined for all real numbers; they had no domain restrictions. However, series do have domain restrictions. A power series may or may not converge for all real values of \(x\text{.}\) The first and most important issue when we start using series as functions is determining the domain of convergence. For power series, we will almost always use the ratio test. Recall that the ratio tests shows convergence when the limit of the ratio of the terms is \(\lt 1\text{.}\) We will use some examples to show the various types of behaviours.

\begin{align*} \sum_{n=0}^\infty \frac{(x+2)^n}{n^2} \amp\\ \lim_{n \rightarrow \infty} \left| \frac{a_{n+1}}{a_n} \right| \amp = \lim_{n \rightarrow \infty} \left| \frac{\frac{(x+2)^{n+1}}{(n+1)^2}}{\frac{(x+2)^n}{n^2}} \right| = \lim_{n \rightarrow \infty} \left| \frac{(x+2)n^2}{(n+1)^2} \right|\\ \amp = |x+2| \lim_{n \rightarrow \infty} \frac{n^2}{n^2+2n+1} = |x+2| \lt 1 \end{align*}

This series is centered at \(\alpha=-2\text{,}\) and the ratio test tells us that we have convergence on \(|x+2|\lt 1\text{,}\) which is the interval \((-3,-1)\text{.}\) Outside the interval, the series diverges and doesn't represent a function. The convergence at the endpoints \(-3\) and \(-1\) is undetermined; we would need to check them individually using another type of test.

\begin{align*} \sum_{n=0}^\infty nx^n \amp\\ \lim_{n \rightarrow \infty} \left| \frac{a_{n+1}}{a_n} \right| \amp = \lim_{n \rightarrow \infty} \left| \frac{x^{n+1} (n+1)}{x^n n} \right| = |x| \lim_{n \rightarrow \infty} \frac{n+1}{n} = |x| \lt 1 \end{align*}

The ratio test allows us to conclude that this converges on \((-1,1)\text{.}\)

\begin{align*} \sum_{n=0}^\infty n!x^n \amp\\ \lim_{n \rightarrow \infty} \left| \frac{a_{n+1}}{a_n} \right| \amp = \lim_{n \rightarrow \infty} \left| \frac{x^{n+1} (n+1)!}{x^n n!} \right| = |x| \lim_{n \rightarrow \infty} \frac{n+1!}{n!} = \infty \end{align*}

This limit is never finite unless \(x=0\text{,}\) so this converges almost nowhere. This is essentially useless as the definition of a function, since its only value is \(f(0) = 0\text{.}\)

\begin{align*} \sum_{n=0}^\infty \frac{(-1)^n (x-7)^n}{2^n n!} \amp\\ \lim_{n \rightarrow \infty} \left| \frac{a_{n+1}}{a_n} \right| \amp = \lim_{n \rightarrow \infty} \left| \frac{\frac{(x-7)^{n+1}}{2^{n+1} (n+1)!}}{\frac{(x-7)^n}{2^n n!}} \right| = |x-7|\lim_{n \rightarrow \infty} \frac{1}{2(n+1)} = 0 \lt 1 \end{align*}

The limit here is \(0\) regardless of the value of \(x\text{,}\) so we have convergence for all real numbers.

The previous examples represent all of the possible types of convergence behaviour of power series. We summarize the situation in a proposition.

Definition 2.1.8.

The positive real number \(R\) in the third case is called the radius of convergence of a power series. We can use this terminology to cover the other two cases as well: in the first case, we say \(R=0\) and in the second case, we say \(R = \infty\text{.}\)

\begin{align*} f(x) \amp = \sum_{n=1}^\infty \sqrt{n} x^n \lim_{n \rightarrow \infty} \frac{a_{n+1}}{a_n} = \lim_{n \rightarrow \infty} \left| \frac{\sqrt{n+1}x^{n+1}}{\sqrt{n} x^n} \right| = \lim_{n \rightarrow \infty} |x| \sqrt{\frac{n+1}{n}}\\ \amp = |x| \lim_{n \rightarrow \infty} \sqrt{ 1 + \frac{1}{n} } = |x| \lt 1 \end{align*}

The radius of convergence is \(R=1\text{,}\) so this series converges on \((-1,1)\text{.}\)

\begin{align*} f(x) \amp = \sum_{n=1}^\infty \frac{n(x-6)^n}{4^{2n+2}} \lim_{n \rightarrow \infty} \frac{a_{n+1}}{a_n} = \lim_{n \rightarrow \infty} \left| \frac{\frac{(n+1)(x-6)^{n+1}}{4^{2n+3}}}{\frac{n(x - 6)^n}{4^{2n+2}}} \right|\\ \amp = |x-6| \lim_{n \rightarrow \infty} \frac{n+1}{n} \frac{1}{4^2} = \frac{|x-6|}{16} \lt 1 \implies |x-6| \lt 16 \end{align*}

The radius of convergence is \(R=16\text{,}\) centered around \(x=6\text{.}\) This series converges on \((-10, 22)\text{.}\)

\begin{align*} f(x) \amp = \sum_{n=1}^\infty \frac{x^n}{7^n} \lim_{n \rightarrow \infty} \frac{a_{n+1}}{a_n} = \lim_{n \rightarrow \infty} \left| \frac{\frac{x^{n+1}}{7^{n+1}}}{\frac{x^n}{7^n}} \right|\\ \amp = |x| \lim_{n \rightarrow \infty} \left| \frac{1}{7} \right| = \frac{|x|}{7} \lt 1 \implies |x| \lt 7 \end{align*}

The radius of convergence is 7 and the series converges on \((-7,7)\text{.}\)

\begin{align*} f(x) \amp = \sum_{n=1}^\infty \frac{x^n}{(1)(3)(5) \ldots (2n+1)} \lim_{n \rightarrow \infty} \frac{a_{n+1}}{a_n} = \lim_{n \rightarrow \infty} \left| \frac{\frac{x^{n+1}}{(1)(3)(5) \ldots (2n+3)}}{\frac{x^n}{(1)(3)(5)\ldots (2n+1)}} \right|\\ \amp = |x| \lim_{n \rightarrow \infty} \frac{1}{2n+3} = 0 \end{align*}

This convergence doesn't depend on \(x\text{,}\) since the limit is \(0\) in any case. Therefore, this has an infinite radius of convergence and is a function defined on all of \(\RR\text{.}\)

Sometimes, we like to simply calculate the radius directly. Here are two formulae to do so.

Subsection 2.1.4 Properties of Power Series

Inside the radius of convergence, a power series has all the properties of a normal function. We can add and subtract two power series as long as we remain inside the radii of both series. We can multiply as well, though the calculations become difficult. The same is true for division: if a series is non-zero inside its radius of convergence, we can divide by the series (though the results of the calculation are difficult to use).

Other properties of series can be calculated with various ease or difficulty, depending on the series. We can investigate the growth of series, whether or not they are bound, symmetric or periodic, and whether or not they are invertible. The key idea to remember is that power series, inside their radii of convergence, are functions; anything that applies to functions can be applied to power series.

Subsection 2.1.5 Calculus of Power Series

Since power series are functions, we can try to do calculus with them, investigating their limits, continuity, derivatives and integrals.

There is a convenient notation for differentiability which we will use frequently.

Definition 2.1.15.

If \(f\) is a function on a domain \(D\) and the \(n\)-th derivative of \(f\) is defined and continuous, we say that \(f\) is in class \(C^n(D)\text{.}\) If the domain is understood implicitly, we just say \(f\) is in class \(C^n\text{.}\) If \(f\) is infinitely differentiable, we say \(f\) is in class \(C^\infty\text{.}\)

The proposition says that power series are in class \(C^\infty\text{,}\) but how are these derivatives calculated? The answer is as nice as possible.

Integration is just as pleasant for power series.

The simplicity of integration is particularly helpful. As we saw in Calculus II, integration is difficult business. For functions which be expressed as series, integration is almost trivial. This makes power series a very useful and convenient class of functions.

Subsection 2.1.6 Series with Patterns of Exponents

If it often the case that a power series may non-zero terms only for certain exponents following some pattern. In this section, I'm going to briefly introduce some common notation for particular cases of this phenomenon. Consider a series where all the odd terms are zero (centered at \(0\) for convenience).

\begin{equation*} f(x) = c_0 + 0x + c_2x^2 + 0x^3 + c_4x^4 + 0x^5 + \ldots \end{equation*}

We could similarly consider a series where all the even terms are zero.

\begin{equation*} f(x) = 0 + c_1x + 0 + c_3x^3 + 0x^4 + c_45^5 + 0x^6 + \ldots \end{equation*}

If we want to index all the even numbers, we can write \(k = 2n\) for \(n \in \NN\text{.}\) Similarly, we can index all the odd numbers by writing \(k = (2n+1)\) for \(n \in \NN\text{.}\) Using these tools, we could write a series with only odd or even non-zero terms. The series

\begin{equation*} f(x) = \sum_{n=0}^\infty c_n (x-\alpha)^{2n} \end{equation*}

is a series with only even terms. The series.

\begin{equation*} g(x) = \sum_{n=0}^\infty c_n (x-\alpha)^{2n+1} \end{equation*}

is a series with only odd terms. When we use these convention, we permit the indices on the \(c_i\) to be unmatched with the exponents. In the function \(g(x)\) above, \(c_1\) matches with \(x^1\text{,}\) but \(c_2\) matches with \(x^3\text{,}\) \(c_3\) matched with \(c^5\) and so on.

Some extra care must be taken with calculating radii of convergence for these series. The formula

\begin{equation*} R = \lim_{n \rightarrow \infty} \left| \frac{c_n}{c_{n+1}} \right| \end{equation*}

relies on the assumption that all \(c_n \neq 0\text{.}\) This is not true for these series with only odd term or even terms. We have to use the ratio test directly to calculate radii of convergence in these situations.

Using similar ideas, we could encode all sorts of patterns in the exponents of our power series. If a power series had non-zero terms only when the exponent was a power of \(3\text{,}\) we could write it as

\begin{equation*} f(x) = \sum_{n=0}^\infty (x-\alpha)^{3n}\text{.} \end{equation*}

If the power series had non-zero exponents only for every fifth number starting at \(7\text{,}\) we could write it as

\begin{equation*} f(x) = \sum_{n=0}^\infty (x-\alpha)^{5n + 7}\text{.} \end{equation*}

Subsection 2.1.7 Non-Elementary Functions

In addition to finding connections between known functions, Taylor series can help us construct entirely new functions. These are often called non-elementary functions (the elementary functions are those which we already have worked with: polynomials, roots, exponentials, logarithms, trig, and hyperbolics).

The Bessel functions of order \(k \in \NN\) are given by this series.

\begin{equation*} J_k(x) = \sum_{n=0}^\infty \frac{(-1)^n x^{2n}}{2^{2n+k} ((n+k)!)^2} \end{equation*}

The Bessel functions are like the trigonometric functions, but the terms in the denominators are larger. They oscilate like trig functions, but with decaying amplitude. They are important for spherical and circular waves, such as sound waves or ripples on a pond.

The Bessel-Clifford functions are givesn by this series.

\begin{equation*} C_k(x) = \sum_{n=0}^\infty \frac{\pi (k+n) x^n}{n!} \end{equation*}

The Polylogarithm functions are given by this series. (Note that for \(s=1\) the polylogarithm is \(Li_1(x) = -\ln (1-x)\text{,}\) the conventional logarithm).

\begin{equation*} Li_s(x) = \sum_{n=0}^\infty \frac{x^{n}}{n^s} \end{equation*}

These three examples are just the very start of a huge world of non-elementary functions.