An infinite series with a constant center \(a\in\mathbb{R}\) and a coefficient sequence \((b_n)_{n\geq 0}\) is called a power series
\[P(x) = \sum\limits_{n=0}^\infty b_n\cdot(x-a)^n\]
In many situations the center of the series \(a=0\), for instance for Maclaurin series. In such cases, the power series takes the simpler form
\[P(x) = \sum\limits_{n=0}^\infty b_n\cdot x^n\]
The question that now arises is, for which values \(x\) does the power series \(P(x)\) converge? Obvious is that if \(x=a\) then
\[\sum\limits_{n=0}^\infty b_n\cdot(x-a)^n = \underbrace{b_0\cdot 0^0}_{b_0} + \underbrace{b_1\cdot 0^1+b_2\cdot 0^2+b_3\cdot 0^3+...}_{0}=b_0\]
converges. In all other cases it depends on the sequence \((b_n)_{n\geq 0}\).
Examples
- \(\sum\limits_{n=0}^\infty \underbrace{1}_{b_n}\cdot (x-\underbrace{0}_a)^n = \sum_{n=0}^\infty x^n\) converges for all \(|x|<1\) (geometric series which is in the convergence interval \(x\in(-1, 1)\)).
- \(\sum\limits_{n=0}^\infty \underbrace{2^n}_{b_n}\cdot (x-\underbrace{0}_a)^n = \sum_{n=0}^\infty (2x)^n\) converges for all \(|2x|<1\), or \(|x|<\frac{1}{2}\), which is in the convergence interval \(x\in\left(-\frac{1}{2}, \frac{1}{2}\right)\).
Radius of Convergence
Every power series has a radius \(R\in[0,\infty]\) such that it converges absolutely for \(|x-a|<R\) and diverges for \(|x-a|>R\). Endpoint behavior at \(|x-a|=R\) must be checked separately.
Proof. Apply the root test to the terms \(|b_n(x-a)^n|\):
\[ \sqrt[n]{|b_n(x-a)^n|}=|x-a|\cdot\sqrt[n]{|b_n|}. \]
Write \(L:=\lim_{n\to\infty}\sqrt[n]{|b_n|}\) (assuming this limit exists). The root test guarantees absolute convergence once this quantity stays below some \(q<1\), which by the computation above happens exactly when
\[ |x-a|\cdot L<1 \quad\Longleftrightarrow\quad |x-a|<\frac1L, \]
and guarantees divergence once \(|x-a|\cdot L>1\), i.e. \(|x-a|>1/L\). So \(R=1/L\) is exactly the threshold between the two behaviors, with the convention \(R=\infty\) if \(L=0\) and \(R=0\) if \(L=\infty\). \(\square\)
Computing the Radius
The proof above identifies the radius directly from the coefficients, the Cauchy–Hadamard formula:
\[ R=\frac{1}{\displaystyle\lim_{n\to\infty}\sqrt[n]{|b_n|}}, \qquad\text{with }\frac10:=\infty. \]
When consecutive coefficients never vanish and the following limit exists, the ratio test gives an often more convenient equivalent formula, sometimes called Euler's formula:
\[ R=\lim_{n\to\infty}\left|\frac{b_n}{b_{n+1}}\right|. \]
Example. For the exponential series \(b_n=\frac{1}{n!}\), the ratio test gives, for any fixed \(x\),
\[ \left|\frac{a_{n+1}}{a_n}\right| = \left|\frac{x^{n+1}/(n+1)!}{x^n/n!}\right| =\frac{|x|}{n+1} \xrightarrow{n\to\infty}0<1 \]
for every real \(x\), so the series converges absolutely no matter how large \(|x|\) is. Equivalently, Euler's formula gives \(R=\lim_{n\to\infty}\left|\frac{1/n!}{1/(n+1)!}\right|=\lim_{n\to\infty}(n+1)=\infty\).
Important Function Expansions
Exponential Function
\[ \exp(x)=\sum\limits_{n=0}^{\infty}\frac{x^n}{n!} =1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\dots \]
With \(b_n=1/n!\), the example above gives \(R=\infty\), so this series converges for every \(x\in\mathbb{R}\) and \(\exp\) is a well-defined function \(\mathbb{R}\to\mathbb{R}\). Setting \(x=0\) leaves only the \(n=0\) term, since \(0^n=0\) for \(n\geq1\):
\[\exp(0)=\frac{0^0}{0!}+\frac{0^1}{1!}+\frac{0^2}{2!}+\cdots=1.\]
The series also satisfies the functional equation
\[\exp(x+y)=\exp(x)\cdot\exp(y)\qquad\forall x,y\in\mathbb{R},\]
obtained by multiplying the two series together and regrouping terms of equal total degree, a Cauchy product. Two useful facts follow directly from it. First, combining \(x\) with \(-x\) and using \(\exp(0)=1\):
\[1=\exp(0)=\exp\bigl(x+(-x)\bigr)=\exp(x)\cdot\exp(-x) \quad\Longrightarrow\quad \exp(-x)=\frac{1}{\exp(x)}.\]
Second, positivity: for \(x\geq0\) every term of the series is nonnegative, so \(\exp(x)\geq1>0\). For \(x<0\), write \(x=-x'\) with \(x'>0\); then \(\exp(x)=\exp(-x')=1/\exp(x')>0\) since \(\exp(x')>0\). Hence
\[\exp(x)>0\qquad\forall x\in\mathbb{R}.\]
Evaluating at \(x=1\) defines Euler's number
\[e:=\exp(1)=\sum_{n=0}^{\infty}\frac1{n!}=\frac1{0!}+\frac1{1!}+\frac1{2!}+\frac1{3!}+\cdots,\]
an irrational number whose partial sums converge quickly:
| \(n\) | \(\sum_{k=0}^{n}1/k!\) |
|---|---|
| 0 | 1 |
| 1 | 2 |
| 2 | 2.5 |
| 3 | 2.666667 |
| 4 | 2.708333 |
| 5 | 2.716667 |
| 6 | 2.718056 |
| 7 | 2.718254 |
approaching \(e\approx2.718281828\ldots\) Applying the functional equation \(n-1\) times gives, for \(n\in\mathbb{N}\),
\[\exp(n)=\exp(\underbrace{1+\cdots+1}_{n\text{ times}})=\underbrace{\exp(1)\cdots\exp(1)}_{n\text{ times}}=e^n,\]
and together with \(\exp(0)=1=e^0\) and \(\exp(-n)=1/\exp(n)=1/e^n=e^{-n}\), this justifies writing \(e^x:=\exp(x)\) for every real \(x\), not just integers.
Sine and Cosine
\[ \sin(x)=\sum\limits_{n=0}^{\infty}\frac{(-1)^n}{(2n+1)!}x^{2n+1}, \]
\[ \cos(x)=\sum\limits_{n=0}^{\infty}\frac{(-1)^n}{(2n)!}x^{2n}. \]