Power series

A power series is an infinite sum built from the powers of a variable: a sum with infinitely many terms, each a constant times a power of x. The general form, centred at a number c, is

\sum_{n=0}^{\infty}a_{n}(x-c)^{n}=a_{0}+a_{1}(x-c)+a_{2}(x-c)^{2}+\cdots

The numbers a₀, a₁, a₂, … are the coefficients, and c is the centre of the series. When c = 0, the powers are simply xⁿ and the series reads a₀ + a₁x + a₂x² + ⋯.

A power series is, in plain words, a polynomial with infinitely many terms. Its usefulness comes from the fact that its terms require only the four basic operations of arithmetic to evaluate: each term a_nxⁿ is a multiplication followed by an addition. Stopping after finitely many terms leaves a polynomial that approximates the value of the whole sum, and keeping more terms improves the approximation (wherever the series converges). Numbers that have no formula built from basic operations alone, such as e^0.05 or sin 1, can therefore be computed to any desired accuracy by arithmetic only. This is the idea behind the scientific functions on calculators and in spreadsheets, and it is the reason power series pervade science.

The geometric series

The simplest power series to study is the one in which every coefficient is 1 and the centre is 0:

1 + x + x² + x³ + ⋯

Each term multiplies the previous one by x; this is called the geometric series. It is the right place to start, because its sum can be worked out exactly, and because it shows clearly when an infinite sum produces a number and when it does not.

Start with the finite version, the partial sum of the first N + 1 terms, S_N = 1 + x + x² + ⋯ + x^N. Multiply both sides of this equation by (1 − x) and multiply out the brackets:

Every middle term cancels: the two brackets share the terms x, x², …, x^N, which subtract to nothing and leave only 1 − x^N+1. Dividing both sides by (1 − x), which is allowed for any x ≠ 1, gives the closed form of the partial sum:

Nothing has been approximated so far: the last formula is exact for every N. The infinite series is defined as the limit of these partial sums as the number of terms grows without bound. Whether such a limit exists depends on where x lies, and the four cases behave very differently.

• Case −1 < x < 1: the series converges. As N grows, the number x^N+1 shrinks towards 0, so the partial sums approach a definite number:

With x = 0.1, for instance, the partial sums run 1; 1.1; 1.11; 1.111; …, settling on 1.111…, and indeed 1/(1 − 0.1) = 1/0.9 = 10/9 = 1.111…

• Case x = 1: the series diverges. Every term equals 1, so S_N = N + 1 grows without bound; the sum 1 + 1 + 1 + ⋯ has no finite value. The formula above cannot be used here, because it divides both sides by 1 − x = 0.

• Case x = −1: the series diverges by oscillation. The partial sums run 1, 0, 1, 0, 1, … and never settle on a single number, so 1 − 1 + 1 − 1 + ⋯ does not converge. The formula 1/(1 − x) would give 1/2 at x = −1, but that value is not the sum of the series; the formula simply does not apply there.

• Case |x| > 1: the series diverges. The terms grow in size like powers of a number bigger than 1, so the partial sums grow without bound. At x = 2 they run 1, 3, 7, 15, …, that is S_N = 2^N+1 − 1.

The geometric series therefore yields a finite number precisely when x lies strictly between −1 and 1. The values of x for which a power series converges form its interval of convergence, which will reappear, in general form, in the next section.

A real-world use: pricing a lifelong income. Suppose an investment grows money by 5% a year. A payment of £10 due one year from now is worth only £10/1.05 today, because £10/1.05 deposited now would grow back to £10 by then; a payment due in two years is worth £10/(1.05)², and so on. A fund that pays £10 every year, forever, with the first payment in one year, is therefore worth today

£10/1.05 + £10/(1.05)² + £10/(1.05)³ + ⋯ = 10r(1 + r + r² + ⋯),

where r = 1/1.05 = 0.952 381…, a number between −1 and 1. The geometric series applies, and since r/(1 − r) = (1/1.05)/(0.05/1.05) = 1/0.05 = 20, the value is 10 × 20 = £200. An infinite number of payments is worth a finite sum today, because payments far in the future are discounted to almost nothing; this is the geometric series in action.

Radius of convergence

The geometric series is one instance of a general fact. For a power series centred at c, convergence depends on how far x lies from the centre. If the series converges for at least one value of x other than c, there is a number R ≥ 0, called the radius of convergence, such that:

• the series converges for every x with |x − c| < R (closer to the centre than R);
• the series diverges for every x with |x − c| > R.

The set |x − c| < R is an interval of length 2R centred at c; it is the interval of convergence of the series. The boundary |x − c| = R is not covered by the statement above: each series must be tested separately at its boundary points, exactly as the geometric series was tested at x = 1 and x = −1 above (there it diverged at both ends, but other series may converge at one or both boundary points).

Where does the radius come from? Compare the size of each term with the one before it. Consecutive terms of a_n(x − c)ⁿ have magnitudes in the ratio |a_n+1/a_n| · |x − c|. If, in the long run, this ratio stays below 1, the series behaves like a convergent geometric series; if it stays above 1, terms grow and the series diverges. The change of behaviour occurs at the value of |x − c| for which the ratio equals 1, namely R = lim |a_n/a_n+1| when this limit exists:

For the geometric series, every coefficient is 1, so the ratio is 1 and R = 1, matching the interval −1 < x < 1 found by direct calculation.

The radius can be any number from 0 to ∞, and the three possibilities behave quite differently:

• Finite radius, R = 1: the geometric series above, and the binomial series below, converge only within a bounded interval around the centre.
• Infinite radius, R = ∞: the series converges for every x. The exponential and sine series in the next section are the standard examples.
• Zero radius, R = 0: the series converges only at the centre itself. For instance the series 1 + x + 2!x² + 3!x³ + ⋯ (whose coefficient of xⁿ is n!) has |a_n/a_n+1| = n!/(n + 1)! = 1/(n + 1) → 0, so R = 0; for any x ≠ 0 the terms eventually grow without bound.

The exponential and sine series

How does one find the power series of a known function such as e^x or sin x? Suppose f can be written as a power series centred at c, with coefficients a_n. Differentiating the series term by term and evaluating at x = c singles out one coefficient at a time, because the n-th derivative of (x − c)ⁿ is the constant n!, and every other term still contains a factor (x − c) that vanishes at x = c. Hence:

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

The coefficients of the series are therefore fixed by the values of f and its derivatives at the centre. The resulting series is called the Taylor series of f about c (about 0 it is also called the Maclaurin series). The two examples below apply this idea to the two most useful functions in science.

The exponential series

The exponential function e^x has the defining property that its derivative equals itself, and e⁰ = 1. Consequently all its derivatives at 0 equal 1: f⁽ⁿ⁾(0) = 1 for every n. The coefficient formula above then gives a_n = 1/n!, so

e^{x}=\sum_{n=0}^{\infty}\frac{x^{n}}{n!}=1+x+\frac{x^{2}}{2!}+\frac{x^{3}}{3!}+\frac{x^{4}}{4!}+\cdots

The factorial n! = 1·2·3·⋯·n in the denominator grows much faster than any power, so the terms shrink quickly and the series converges for every x. Applying the ratio test confirms R = ∞:

Setting x = 1 in the series gives the number e itself: e = 1 + 1 + 1/2 + 1/6 + 1/24 + ⋯ = 2.718 281 828…

Numeric example: compound interest. Suppose £1000 sits in an account that pays 5% interest per year, compounded continuously (interest credited at every instant and immediately earning interest itself). After one year the balance is £1000 · e^0.05. The series computes this number with arithmetic alone; taking x = 0.05:

Partial sums of e^0.05, and the resulting balance

Terms included	Partial sum	Balance of £1000
1	1	£1000.00
1 + 0.05	1.05	£1050.00
+ 0.05²/2!	1.05125	£1051.25
+ 0.05³/3!	1.051 2708	£1051.27
+ 0.05⁴/4!	1.051 2711	£1051.27

After the fourth term the balance has already settled to the nearest penny: the account holds about £1051.27 after a year, an effective annual rate of 5.127%.

The sine series

The derivatives of sin x cycle through sin x, cos x, −sin x, −cos x, and back. Evaluated at 0 they cycle through 0, 1, 0, −1, so only the odd powers survive, with alternating signs and coefficient sizes 1, 1/3!, 1/5!, …. Hence:

\sin x=x-\frac{x^{3}}{3!}+\frac{x^{5}}{5!}-\frac{x^{7}}{7!}+\frac{x^{9}}{9!}-\cdots

The sine series also converges for every x: each new term is the previous one multiplied by x² and divided by two ever-larger factors, so the multiplier tends to 0 no matter how large x is:

Numeric example: how high a tilted plank rises. The sine of an angle in a right triangle is the ratio of the opposite side to the hypotenuse. A straight plank 1 m long, tilted up at an angle of 1 radian (about 57.3°), therefore rises sin 1 metres above the ground. The series with x = 1 gives, adding terms one by one,

Partial sums of sin 1

Terms included	Partial sum
1	1
1 − 1/6	0.833 333
+ 1/120	0.841 667
− 1/5040	0.841 468
+ 1/362 880	0.841 471

The sums settle on 0.841 471, which is sin 1 correct to six decimal places. A plank 10 m long at the same angle rises about 8.41 m; a seat on a Ferris wheel of radius 10 m, having turned through 1 radian from the level of the hub, sits about 8.41 m above that level.

The binomial series

One more family of power series is useful enough to know by name. For any fixed exponent p,

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

Two cases behave differently and should not be confused:

• p a non-negative whole number (0, 1, 2, …): the series stops by itself, because the factor (p − p) appears after p + 1 terms. It is the ordinary binomial expansion of a polynomial, for example (1 + x)² = 1 + 2x + x².
• Any other exponent p, a fraction, a negative number, an irrational number: the series never stops, and it is a genuine infinite series valid for −1 < x < 1 (radius R = 1). The geometric series of the second section is the case p = −1 with x replaced by −x, since 1/(1 + x) = (1 + x)⁻¹ = 1 − x + x² − x³ + ⋯.

Numeric example: a square root from arithmetic. Put p = 1/2, so that (1 + x)^1/2 is just √(1 + x):

Take x = 0.1. Then √1.1 = 1 + 0.05 − 0.00125 + 0.000 0625 − 0.000 0039 + ⋯ = 1.048 8086…, whereas √1.1 = 1.048 8088…; the first few terms already give six correct decimal places. A pendulum of length L swings with period 2π√(L/g), where g is the acceleration of gravity, so lengthening a pendulum by 10% multiplies its period by √1.1 ≈ 1.0488: each swing takes about 4.9% longer.

Power series solutions of differential equations

A deeper application of power series is to differential equations whose coefficients vary with x. Such equations rarely have solutions built from a finite combination of familiar functions, but when the coefficient functions can be expanded in power series near a point, a solution can still be written down as a power series about that point. Substituting the series into the equation and equating the coefficients of like powers of x turns the differential equation into recurrence relations that fix the coefficients one after another; the constants left free by the recurrence are exactly the arbitrary constants of the equation, and the resulting series solves the equation exactly on its interval of convergence. The place of series among the approaches to differential equations is discussed in the article Differential equation.

Case 1: expanding about an ordinary point, the Airy equation

The Airy equation

y″ − xy = 0

is named after George Biddell Airy, who met it in 1838 while studying the intensity of light near a caustic. Its solutions cannot be written as finite combinations of elementary functions, so it is the standard test case for series methods.

Near x = 0 the coefficients of the equation are as well behaved as possible (the point is an ordinary point), so seek a solution as a power series about 0: y = a₀ + a₁x + a₂x² + ⋯, where the a_n are unknown constants. Differentiate term by term: y′ = a₁ + 2a₂x + 3a₃x² + ⋯ and y″ = 2a₂ + 6a₃x + 12a₄x² + ⋯, so each derivative of xⁿ lowers the power by one and multiplies the coefficient by n. Re-indexing the sums so that both run over the same powers of x gives

Substituting these two expressions into y″ − xy = 0 and collecting terms, the equality must hold for every value of x; that forces the coefficient of each power of x on the left to vanish. The constant term gives 2·1·a₂ = 0, so a₂ = 0, and comparing the coefficient of xⁿ for n ≥ 1 gives the recurrence

Each new coefficient is fixed by the one three places earlier. Writing the recurrence out, starting at n = 1, 2, 3, …:

a₃ = a₀/(3·2) = a₀/6,　 a₆ = a₃/(6·5) = a₀/180,

a₄ = a₁/(4·3) = a₁/12,　 a₇ = a₄/(7·6) = a₁/504,

a₅ = a₂/(5·4) = 0,　 a₈ = a₅/(8·7) = 0,

and so on. The coefficients split into three chains: those with indices divisible by 3 are fixed by a₀, those one more than a multiple of 3 are fixed by a₁, and those two more than a multiple of 3 are all zero (they start from a₂ = 0). The two free constants a₀ and a₁, the arbitrary constants expected of a second-order equation, generate two independent solutions:

Each series converges for every x (the denominators grow like factorials, so the ratio test gives R = ∞). Up to normalisation these two solutions are the Airy functions Ai and Bi:

y''-xy=0\qquad\Longrightarrow\qquad y=C_{1}\operatorname{Ai}(x)+C_{2}\operatorname{Bi}(x)

The power series therefore solve the Airy equation exactly, in open form, on the whole real line. The Airy functions appear in optics, quantum mechanics and elsewhere whenever a solution changes from oscillatory to exponential behaviour.

Case 2: expanding about a singular point, the Frobenius method

Write a linear second-order equation in the form y″ + p(x)y′ + q(x)y = 0. A point x₀ is ordinary when both p and q can be expanded in power series about x₀; the direct method of Case 1 applies there. If at least one of p or q cannot, typically because it divides by x − x₀, the point is singular. The point is a regular singular point when (x − x₀)p(x) and (x − x₀)²q(x) do have power series expansions about x₀.

At a singular point a plain power series cannot work: with p or q dividing by x, the equation mixes powers whose lowest terms start at different heights, so they cannot be matched. The Frobenius method instead seeks a solution with a shifted leading power, y = x^r(a₀ + a₁x + ⋯). Substituting and setting the coefficient of the lowest power of x to zero produces an algebraic equation, the indicial equation, whose solutions fix the possible exponents r. For the Bessel equation of order ν,

x^{2}y''+x\,y'+(x^{2}-\nu^{2})\,y=0;\qquad r^{2}-\nu^{2}=0\ \Rightarrow\ r=\pm\nu;\qquad y=C_{1}J_{\nu}(x)+C_{2}Y_{\nu}(x)

the indicial equation is r² − ν² = 0, with roots r = ±ν. When ν is not a whole number the two roots give two independent series solutions, the Bessel functions of the first kind, J_ν and J_−ν. When the roots coincide or differ by a whole number, the second independent solution acquires a logarithmic term; for ν a whole number this second solution is the Bessel function of the second kind Y_ν. Bessel functions describe vibrations of a circular drumhead, heat flow in a cylinder and waves in optical fibres. The full theory is developed in standard textbooks on ordinary differential equations.

Further reading

• Power series, Wikipedia
• Power Series, Wolfram MathWorld
• Frobenius method, Wikipedia
• Power Series, Paul's Online Math Notes