6.1 Power Series and Functions

Power series let you represent functions as infinite sums of polynomial-like terms. This is one of the most powerful tools in Calculus II because it connects infinite series to actual functions you can differentiate, integrate, and use to approximate values.

Power Series

Construction of power series

A power series is an infinite series of the form:

$\sum_{n=0}^{\infty} a_n(x-c)^n = a_0 + a_1(x-c) + a_2(x-c)^2 + a_3(x-c)^3 + \cdots$

where $a_n$ are the coefficients, $c$ is the center, and $x$ is the variable. Think of it as an "infinite polynomial" built around the point $c$.

Two important special cases:

• Maclaurin series: a power series centered at $c = 0$. For example, $e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!}$.
• Taylor series: a power series centered at any point $c$, with the general form $\sum_{n=0}^{\infty} \frac{f^{(n)}(c)}{n!}(x-c)^n$. Every Maclaurin series is a Taylor series, just one where $c = 0$.

You can generate new power series from known ones by substituting expressions for the variable. For example, if you know the Maclaurin series for $e^x$, you can find the series for $e^{x^2}$ by replacing every $x$ with $x^2$:

$e^{x^2} = \sum_{n=0}^{\infty} \frac{(x^2)^n}{n!} = \sum_{n=0}^{\infty} \frac{x^{2n}}{n!}$

Radius of convergence calculation

A power series won't converge for every value of $x$. The radius of convergence $R$ tells you how far from the center $c$ the series converges.

The most common way to find $R$ is the Ratio Test:

1. Compute $L = \lim_{n\to\infty} \left|\frac{a_{n+1}}{a_n}\right|$, where $a_n$ is the coefficient of $(x-c)^n$.
2. The radius of convergence is $R = \frac{1}{L}$.
   • If $L = 0$, then $R = \infty$ (the series converges for all $x$).
   • If $L = \infty$, then $R = 0$ (the series converges only at $x = c$).

The interval of convergence is the set of all $x$-values where the series converges. Finding it requires two steps:

1. Use the radius to get the open interval $(c - R,\; c + R)$.

2. Check the endpoints $x = c - R$ and $x = c + R$ individually, since the Ratio Test is inconclusive there. You'll need other tests (like the Alternating Series Test or $p$-series comparison) at the endpoints.

Example: For $\sum_{n=1}^{\infty} \frac{x^n}{n}$, the Ratio Test gives $R = 1$, so the open interval is $(-1, 1)$. At $x = 1$, you get the harmonic series $\sum \frac{1}{n}$, which diverges. At $x = -1$, you get the alternating harmonic series $\sum \frac{(-1)^n}{n}$, which converges. So the interval of convergence is $[-1, 1)$.

Functions as power series representations

The Taylor series expansion lets you write a function as a power series:

$f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(c)}{n!}(x-c)^n$

You compute the derivatives of $f$ at the center $c$, plug them in as coefficients, and get a series that equals $f(x)$ within the interval of convergence.

Why this matters:

• Approximating function values: Truncate the series after a few terms to estimate values near the center. For instance, using the first few terms of the Maclaurin series for $\sin(x)$, you can approximate $\sin(0.1) \approx 0.1 - \frac{(0.1)^3}{6} = 0.09983\ldots$
• Solving differential equations: Some differential equations (like $y'' - xy = 0$) don't have solutions in terms of elementary functions, but you can find power series solutions.
• Defining functions precisely: The series $\sum_{n=0}^{\infty} \frac{(-1)^n}{(2n+1)!}x^{2n+1}$ is the definition of $\sin(x)$ in a rigorous sense.

Functions as Power Series

Functions as power series representations

Several standard power series show up constantly. Memorize these:

• Geometric series: $\frac{1}{1-x} = \sum_{n=0}^{\infty} x^n$ for $|x| < 1$
• Exponential series: $e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!}$ for all $x$
• Binomial series: $(1+x)^r = \sum_{n=0}^{\infty} \binom{r}{n}x^n$ for $|x| < 1$, where $\binom{r}{n} = \frac{r(r-1)(r-2)\cdots(r-n+1)}{n!}$

You can build many other series from these using substitution. For example, replacing $x$ with $-x^2$ in the geometric series gives:

$\frac{1}{1+x^2} = \sum_{n=0}^{\infty} (-1)^n x^{2n} \quad \text{for } |x| < 1$

Operations on power series work much like operations on polynomials, but with infinitely many terms:

• Addition/subtraction: Add or subtract corresponding coefficients. For $\cos(x) + \sin(x)$, just add the two known series term by term.
• Multiplication: Multiply series and collect like powers of $x$. This is called the Cauchy product.
• Term-by-term differentiation: If $f(x) = \sum a_n x^n$, then $f'(x) = \sum n \cdot a_n x^{n-1}$. The radius of convergence stays the same (though endpoint behavior may change).
• Term-by-term integration: $\int f(x)\,dx = C + \sum \frac{a_n}{n+1} x^{n+1}$. Again, the radius of convergence is preserved.

These operations are especially useful for:

• Evaluating tricky limits: Expand the functions as series and simplify. For example, $\lim_{x\to 0} \frac{e^x - 1}{x}$: replace $e^x$ with $1 + x + \frac{x^2}{2} + \cdots$, subtract 1, divide by $x$, and you get $1 + \frac{x}{2} + \cdots \to 1$.
• Approximating integrals: If a function has no elementary antiderivative (like $e^{-x^2}$), you can integrate its power series term by term to get a series for the integral, then sum enough terms for the accuracy you need.

Convergence and Divergence of Power Series

Every power series falls into one of three convergence cases:

1. Converges only at $x = c$ (radius $R = 0$)
2. Converges for all $x$ (radius $R = \infty$)
3. Converges on some finite interval around $c$ (radius $R > 0$)

Two types of convergence to distinguish:

• Absolute convergence: The series $\sum |a_n(x-c)^n|$ converges. Inside the open interval $(c-R, c+R)$, a power series always converges absolutely.
• Conditional convergence: The series converges, but the series of absolute values diverges. This can only happen at the endpoints of the interval of convergence.

Outside the radius of convergence, the series always diverges.

Analytic Functions and Laurent Series

An analytic function is one that equals its own Taylor series in some neighborhood of each point in its domain. Within the radius of convergence, every power series defines an analytic function. A key property: if two power series centered at the same point converge to the same function, their coefficients must be identical. This uniqueness is what makes power series representations so useful.

Laurent series generalize power series by allowing negative powers of $(x - c)$:

$\sum_{n=-\infty}^{\infty} a_n(x-c)^n$

These are primarily a topic in complex analysis, but the basic idea is that Laurent series can represent functions near points where the function blows up (singularities), which ordinary Taylor series can't handle. If you encounter Laurent series in Calc II, it's likely just as a preview of what's ahead.