Power Series

Every time your phone works out a sine, a logarithm or a square root, there is no tiny protractor hidden in the chip — it quietly adds up a never-ending polynomial to reach the answer. That never-ending polynomial is a power series, and it is how machines compute almost every function that science and engineering run on.

What if a polynomial never had to stop? Polynomials are the friendliest functions in mathematics — you can evaluate them with nothing but multiplication and addition, differentiate them in your head, integrate them by shifting an exponent. Their only flaw is that they run out: a degree-7 polynomial has nothing to say after its x^7 term. A power series removes the stopping point. Pick a centre a and an endless supply of coefficients c_0, c_1, c_2, \dots, and form

f(x) \ = \ \sum_{n=0}^{\infty} c_n (x - a)^n \ = \ c_0 + c_1(x-a) + c_2(x-a)^2 + \cdots.

This is a polynomial of infinite degree. For each fixed x it is just a numerical series — it either converges or it doesn't — and the set of x where it converges is the domain of the function f. The astonishing payoff, which this page unpacks, is that wherever it converges it behaves exactly like a polynomial: you may differentiate it, integrate it, and add it term by term. Suddenly the traffic flows both ways — functions become series (we can compute \ln and \arctan by adding up powers of x) and series become functions (an infinite sum acquires a name, a graph, a derivative). Most of the functions your calculator knows are, under the hood, exactly this.

The mother of all power series

Take a = 0 and every coefficient c_n = 1. The power series is the geometric series in x:

\sum_{n=0}^{\infty} x^n = 1 + x + x^2 + x^3 + \cdots.

Step 1 — the partial sum has a closed form. Multiply S_N = 1 + x + \cdots + x^N by (1-x); the middle terms telescope:

(1-x)S_N = 1 - x^{N+1} \quad\Longrightarrow\quad S_N = \frac{1 - x^{N+1}}{1 - x} \quad (x \ne 1).

Step 2 — take the limit. If |x| < 1 then x^{N+1} \to 0, so S_N \to 1/(1-x). If |x| \ge 1 the terms x^n do not even tend to zero and the series diverges. Hence

\sum_{n=0}^{\infty} x^n = \frac{1}{1 - x}, \qquad \text{valid exactly on } (-1, 1).

A simple-looking sum on the left equals a genuine, named function on the right — but only inside (-1,1). Where a power series converges is not optional; it is part of what the function is. We call this the mother series because, as you are about to see, an entire nursery of famous series can be bred from it without ever doing another limit computation.

Calculus, term by term

The real magic is that, inside its interval of convergence, a power series may be differentiated and integrated one term at a time, just like a polynomial — and the new series has the same radius. Differentiate the geometric series:

\frac{d}{dx}\sum_{n=0}^{\infty} x^n = \sum_{n=1}^{\infty} n x^{n-1} = \frac{d}{dx}\,\frac{1}{1-x} = \frac{1}{(1-x)^2}.

We never touched the limit machinery — we differentiated the closed form on the right and the series on the left, and they agree. Likewise integrating term by term gives a new power series for -\ln(1-x):

\int_0^x \frac{dt}{1-t} = \sum_{n=0}^{\infty} \frac{x^{n+1}}{n+1} = x + \frac{x^2}{2} + \frac{x^3}{3} + \cdots = -\ln(1-x).

One geometric series, differentiated and integrated, has just handed us power series for 1/(1-x)^2 and \ln. That is the engine of the whole subject.

A power series f(x) = \sum_{n=0}^{\infty} c_n (x-a)^n defines a function on the set of x where it converges. On the open interval where it converges:

A single convergent power series is automatically C^\infty on the interior of its interval — you can differentiate it forever and every derivative is again a power series. This is a small miracle. An arbitrary function built by other means might be differentiable twice and then break; a power series never does. Each differentiation lowers every exponent by one and multiplies by it, so

f^{(k)}(x) = \sum_{n=k}^{\infty} n(n-1)\cdots(n-k+1)\, c_n (x-a)^{n-k},

which is still a power series with the same radius. The catch — and it is a sharp one — is the phrase "inside the interval". The convergence is what licenses every interchange of limit, derivative and integral above. Step outside the interval and the series is not a function at all, merely a divergent symbol. The next page, radius of convergence, is devoted to measuring exactly how wide that safe interval is.

Watch the series become the function

The faint curve is the true function 1/(1-x). The bold curve is the truncated power series \sum_{n=0}^{N} x^n — an honest polynomial of degree N. Drag N upward and watch two things at once. Inside (-1,1) the polynomial snaps onto the function — near x = 0 the fit is superb even for small N, because there the discarded terms x^{N+1} + x^{N+2} + \cdots are microscopic. But near the boundary x = \pm 1 the polynomial flails: at x = -1 it flips between 0 and 1 forever, and left of -1 it shoots off in whichever direction its last term points. No finite N ever tames the region |x| \ge 1 — that is the boundary of convergence making itself felt, and no amount of patience crosses it.

A factory for new series

Here is the trade secret of every mathematician who ever wrote down a famous series: almost none of them were derived from scratch. They were manufactured from the mother series by two moves — substitution and term-by-term calculus.

Move 1: substitute. The identity \sum u^n = 1/(1-u) holds for any expression u with |u| < 1. Feed it u = -x^2:

\frac{1}{1+x^2} = \frac{1}{1-(-x^2)} = \sum_{n=0}^{\infty} (-x^2)^n = 1 - x^2 + x^4 - x^6 + \cdots,

valid when |-x^2| < 1, i.e. on (-1,1) again. One line of algebra, no limits, and we own a power series for a function that looks nothing like a geometric sum. (Try u = -x and u = 2x yourself: you should get 1/(1+x) on (-1,1) and 1/(1-2x) on |x| < \tfrac12 — note how the substitution rescaled the interval.)

Move 2: integrate term by term. Now let us derive the series for \ln(1+x) honestly, every step in view. Start from the substitution u = -t:

\frac{1}{1+t} = \sum_{n=0}^{\infty} (-1)^n t^n = 1 - t + t^2 - t^3 + \cdots, \qquad |t| < 1.

Integrate both sides from 0 to x (with |x| < 1, so the whole path of integration stays inside the safe interval). The left side is a calculus classic; the right side goes term by term, each t^n becoming x^{n+1}/(n+1):

\ln(1+x) \;=\; \int_0^x \frac{dt}{1+t} \;=\; \sum_{n=0}^{\infty} (-1)^n \frac{x^{n+1}}{n+1} \;=\; x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \cdots.

Both constants of integration vanish because both sides are 0 at x = 0. The same recipe applied to our 1/(1+t^2) series delivers, with no extra effort, one of the most celebrated formulas in mathematics:

\arctan x \;=\; \int_0^x \frac{dt}{1+t^2} \;=\; x - \frac{x^3}{3} + \frac{x^5}{5} - \frac{x^7}{7} + \cdots, \qquad |x| < 1.

Pause on what just happened. The logarithm and the inverse tangent — transcendental functions, unreachable by any finite amount of school algebra — have been written as limits of polynomials, by doing nothing more than substituting into a geometric series and integrating powers of t. Functions became series.

Before calculus was even public, the young Isaac Newton had discovered that (1+x)^p has a power series for any exponent p — not just whole numbers. His generalised binomial series,

(1+x)^{p} = 1 + px + \frac{p(p-1)}{2!}x^2 + \frac{p(p-1)(p-2)}{3!}x^3 + \cdots \qquad (|x|<1),

turns square roots into sums: with p = \tfrac12 it gives \sqrt{1+x} = 1 + \tfrac{x}{2} - \tfrac{x^2}{8} + \tfrac{x^3}{16} - \cdots. Newton treated it as a private superpower — he used such series to compute areas, roots and the digits of \pi to extraordinary accuracy, and confessed he was "ashamed" to admit how many digits he once computed, "having no other business at the time". When rivals pressed him for his methods, he famously encoded one claim about series as an anagram. To Newton, every function worth having was a power series in disguise — and the pages ahead, on Taylor series, largely prove him right.

The factory's product, on screen

Here is the arctangent series earning its keep. The faint curve is the true \arctan x; the bold one is the polynomial x - \frac{x^3}{3} + \cdots \pm \frac{x^{2M+1}}{2M+1}. Inside (-1,1) a handful of terms already hugs the curve. But look just past x = \pm 1: the polynomial peels away violently, no matter how large you push M. The series inherited its interval from the geometric series it was built from, and integrating term by term did not widen it.

Cashing in: evaluating a series at a point

A power series is not just theory — it is a computational device. Any x inside the interval may be plugged in, and the partial sums then converge to a genuine number. Let us compute \ln(1.1) by setting x = 0.1 (comfortably inside (-1,1)) in our freshly built series:

\ln(1.1) = 0.1 - \frac{(0.1)^2}{2} + \frac{(0.1)^3}{3} - \frac{(0.1)^4}{4} + \cdots = 0.1 - 0.005 + 0.000333\ldots - 0.000025 + \cdots

Three terms give 0.095333\ldots; four give 0.0953083\ldots; the true value is \ln(1.1) = 0.0953102\ldots Each extra term buys roughly another digit, because the powers of 0.1 collapse so fast. This is the general pattern: the deeper inside the interval you are, the faster the series converges. At x = 0.9 the same series still converges — but it crawls, and near the edge of the interval it crawls painfully. A power series is a machine whose efficiency depends on where you stand.

Changing the centre

So far every series has been centred at 0, which is useless if the function you care about misbehaves there — 1/x, say, which blows up at the origin. The fix is to expand in powers of (x-a) about a friendlier centre. Take a = 1 and massage 1/x until a geometric series appears:

\frac{1}{x} = \frac{1}{1 + (x-1)} = \sum_{n=0}^{\infty} (-1)^n (x-1)^n = 1 - (x-1) + (x-1)^2 - (x-1)^3 + \cdots,

valid when |x - 1| < 1, i.e. on the interval (0, 2). Notice two things. First, the interval is symmetric about the centre a = 1, not about 0 — a power series always converges on a symmetric interval (a - R,\, a + R) around its centre (plus possibly an endpoint or two), where R is its radius of convergence. Second, look where the interval stops: exactly at x = 0, the point where 1/x explodes. The series seems to know about the singularity and refuses to reach it — the radius here is precisely the distance from the centre to the nearest catastrophe. Measuring R in general (it can be 0, a finite number, or \infty) is the business of the next page.

Every manipulation on this page is licensed only inside the interval of convergence. Forget that, and algebra becomes a nonsense generator. Plug x = 2 into \sum x^n = 1/(1-x) and you "prove"

1 + 2 + 4 + 8 + 16 + \cdots = \frac{1}{1-2} = -1,

an ever-growing sum of positive numbers equalling -1. Nothing in the algebra failed — the closed form S_N = (1-x^{N+1})/(1-x) is perfectly true at x=2. What failed is the limit: for |x| \ge 1 the term x^{N+1} never dies, so the series diverges and the equation with 1/(1-x) simply does not apply. Divergent series obey no arithmetic.

A subtler trap: term-by-term calculus preserves the radius but can change what happens at the endpoints. The geometric series diverges at both x = 1 and x = -1. Integrate it and you get x + \frac{x^2}{2} + \frac{x^3}{3} + \cdots = -\ln(1-x): still radius 1, still divergent at x = 1 (the harmonic series), but at x = -1 it now converges — it is the alternating harmonic series, summing to -\ln 2. Integration soothed an endpoint into convergence; differentiation can do the reverse and destroy one. Moral: the open interval travels with the series for free, but every endpoint must be re-checked by hand.

Press sin on a calculator and something has to happen inside a chip that can only add, subtract, multiply and divide. There is no tiny protractor in there. What the chip evaluates is — a truncated power series (or a polynomial fine-tuned from one). Functions like \sin, \cos and e^x all have power series converging for every x, and near 0 a handful of terms is spectacular: for |x| \le 0.5, x - \frac{x^3}{6} + \frac{x^5}{120} already matches \sin x to about eight decimal places. So the software first uses identities to fold your input into a small interval around 0 (where convergence is fastest — exactly the "deeper inside, faster" principle above), then fires off a short polynomial. Your slider experiments on this page are a faithful cartoon of what happens on every keypress.

A lovely historical aside: many pocket calculators and early flight computers used a different trick called CORDIC, which computes \sin and \cos purely by shifting binary digits and adding — no multiplier circuit needed. But modern processors, with fast multipliers on board, have gone back to polynomials. Three and a half centuries after Newton, the state of the art is still: truncate a power series.

See it explained