Difficulty: Intermediate | Prerequisites: Geometric series, ratio test, p-series test, absolute vs conditional convergence.
Power series are the bridge between polynomials and infinite series. They let you represent functions as infinitely long polynomials, which turns out to be enormously useful in calculus and engineering. This topic builds directly on geometric series and convergence tests from earlier in the course, so make sure those are solid before diving in. If you can sum a geometric series and apply the ratio test, you have what you need to start here.
A power series is an infinite polynomial in x (or in x minus some constant a). It only "works" for certain values of x, and the set of those values is always an interval, called the interval of convergence. You find that interval using the ratio test, then check the endpoints separately.
Power series
A function defined as an infinite sum of terms involving powers of x (or powers of x minus a constant). Formally: f(x) = the sum from n = 0 to infinity of c_n times x^n. In simple terms, think of it as a polynomial that never stops: it just keeps adding terms with higher and higher powers of x.
Coefficient (c_n)
The constant multiplier attached to each term in the power series. In the sum of c_n times x^n, c_n is the nth coefficient. Think of it as the "weight" each power of x carries in the series.
Centre (a)
The value around which a general power series is built. In the sum of c_n times (x minus a)^n, the constant a is the centre. In simple terms, it is the point the series "orbits around." When a = 0, you get the basic form with plain powers of x.
Interval of convergence (I.O.C.)
The set of all x-values for which a power series converges to a finite number. For any power series, this set is always an interval. Think of it as the "safe zone" where the series actually produces a meaningful answer.
Radius of convergence (R)
Half the width of the interval of convergence. For a series centred at a, the I.O.C. has the form (a minus R, a plus R), possibly including one or both endpoints. In simple terms, R tells you how far from the centre you can go before the series blows up.
Domain (of a power series)
The set of x-values where the power series is defined and converges. This is identical to the interval of convergence, including whichever endpoints work. Think of it as the answer to "for which values of x does this actually equal something finite?"
The geometric series is the starting point. Recall that the sum from n = 0 to infinity of r^n equals 1/(1 minus r) when |r| < 1, and diverges otherwise.
Replace r with x, and you get a function:
f(x) = sum from n = 0 to infinity of x^n = 1 + x + x^2 + x^3 + ...
This equals 1/(1 minus x) when |x| < 1
The domain is the open interval (minus 1, 1)
This is already a power series, the simplest one. It converges inside its interval and diverges outside it.
Worked example: evaluating f(x) = sum of x^n
f(1/2) = 1 + 1/2 + 1/4 + 1/8 + ... = 1/(1 minus 1/2) = 2. This works because 1/2 is inside (minus 1, 1).
f(minus 1/3) = 1 minus 1/3 + 1/9 minus 1/27 + ... = 1/(1 minus (minus 1/3)) = 3/4. Also inside the domain.
f(0) = 1 + 0 + 0 + 0 + ... = 1/(1 minus 0) = 1.
f(5) = 1 + 5 + 25 + 125 + ... = undefined (diverges). The value 5 is outside the domain.
Consider f(x) = sum from n = 0 to infinity of (minus 1)^n times x^n / 3^n.
This can be rewritten as the sum of (minus x/3)^n, which is a geometric series with ratio minus x/3.
It converges when |minus x/3| < 1, i.e. |x/3| < 1, i.e. |x| < 3.
The domain is the open interval (minus 3, 3).
On this interval, the series equals 1/(1 minus (minus x/3)) = 3/(3 + x).
A power series (centred at 0) is a function of the form:
f(x) = sum from n = 0 to infinity of c_n times x^n = c_0 + c_1 x + c_2 x^2 + c_3 x^3 + ...
Here c_n is the nth coefficient. In the first example above, every c_n = 1. In the second example, c_n = (minus 1)^n / 3^n.
More generally, a power series centred at x = a is:
f(x) = sum from n = 0 to infinity of c_n times (x minus a)^n = c_0 + c_1 (x minus a) + c_2 (x minus a)^2 + ...
The constant a shifts the "anchor point" of the series. When a = 0, this reduces to the basic form.
We say this series is "centred at a," or "about a," or "around a." All three phrasings mean the same thing.
By definition, a power series uses whole-number (non-negative integer) powers of x. An expression like the sum of (1/n) times x^(n/2) involves x^(1/2), x^(3/2), etc., which are not whole-number powers. That is not a power series.
Geometric series (the foundation)
Sum from n = 0 to infinity of r^n = 1/(1 minus r), valid when |r| < 1.
Basic power series (centred at 0)
f(x) = sum from n = 0 to infinity of c_n times x^n
General power series (centred at a)
f(x) = sum from n = 0 to infinity of c_n times (x minus a)^n
Interval of convergence structure
For a series centred at a with radius of convergence R, the I.O.C. takes the form (a minus R, a plus R), possibly including one or both endpoints. The centre a is always the midpoint of the interval.
Power series are how calculators and computers evaluate functions like sin(x), cos(x), e^x, and ln(x). Your calculator does not "know" what sin(0.5) is; it sums enough terms of the Taylor series (a type of power series) to get the answer to the required precision. Engineers use power series to approximate complicated functions with simpler polynomials, which is essential in signal processing, control systems, and numerical methods.
Students often think a power series converges everywhere or nowhere. It does not. Every power series has a specific interval of convergence, and it may converge at some points and diverge at others.
Students sometimes confuse the centre a with the radius R. The centre is where the interval sits on the number line. The radius is how wide it stretches from that centre.
Fractional or negative powers of x do not produce a power series. The sum of x^(n/2) is not a power series, even though it looks similar. Power series require whole-number exponents (0, 1, 2, 3, ...).
A common error is forgetting to check the endpoints of the interval of convergence. The ratio test only finds the open interval. The endpoints must be tested individually using other convergence tests.
⚠️ Expect to be given a power series and asked to identify the centre, the coefficients, and the interval of convergence. This is bread-and-butter exam material.
⚠️ Know the geometric series formula cold. Many exam questions start by recognising a given series as a geometric series in disguise.
⚠️ Be ready to evaluate a power series at a specific x-value by recognising the closed-form function it represents (e.g. 1/(1 minus x) for the sum of x^n).
⚠️ "Write the first four terms" is a common prompt. Expand the series by plugging in n = 0, 1, 2, 3.
True or false: The sum from n = 0 to infinity of x^n converges for all real numbers x. (False. It converges only for |x| < 1.)
Fill in the blank: A power series centred at a = 3 with radius of convergence R = 5 has an interval of convergence of the form ( ___ , ___ ), before checking endpoints. (Answer: (minus 2, 8).)
True or false: The expression sum of x^(n/2) from n = 0 to infinity is a power series. (False. It involves fractional powers of x.)
Fill in the blank: In the power series sum of c_n times x^n, the values c_n are called the ___. (Answer: coefficients.)
True or false: If a geometric series has ratio x/3, it converges when |x| < 3. (True.)
Q: Write the first four non-zero terms of the power series f(x) = sum from n = 0 to infinity of (2x)^n. What is the domain of this series?
A: The first four terms are 1 + 2x + 4x^2 + 8x^3. This is a geometric series with ratio 2x. It converges when |2x| < 1, i.e. |x| < 1/2. The domain is (minus 1/2, 1/2).
Q: A power series is centred at a = 4. Its interval of convergence is (1, 7). What is the radius of convergence?
A: R = 3. The interval stretches 3 units in each direction from the centre a = 4.
Q: Is the series sum from n = 0 to infinity of (minus 1)^n times x^n / 3^n a power series? If so, identify the coefficients c_n and find the domain.
A: Yes, it is a power series with c_n = (minus 1)^n / 3^n. It can be rewritten as the geometric series with ratio (minus x/3), so it converges when |x/3| < 1, giving domain (minus 3, 3). On this interval it equals 3/(3 + x).
Q: Explain why sum from n = 1 to infinity of n times x^(1/n) is not a power series.
A: A power series requires whole-number (non-negative integer) powers of x. The exponent 1/n produces fractional powers (x^1, x^(1/2), x^(1/3), ...), so this expression does not fit the definition.
This connects directly to Taylor and Maclaurin series, which are power series whose coefficients are determined by the derivatives of a known function. Once you understand power series, Taylor series are a natural next step.
The convergence tests you learned earlier in the course (ratio test, p-series test, alternating series test) all come back here when you need to determine the interval of convergence and check endpoints.
Power series also connect to differential equations: many differential equations that cannot be solved with elementary functions can be solved by assuming the solution is a power series and finding its coefficients.
Power series, infinite polynomial, geometric series, convergence, divergence, interval of convergence, I.O.C., radius of convergence, centre of a power series, series centred at a, domain of a series, coefficients, c_n, Calculus II, Calculus 2, Calc 2, XX123, series and sequences, Taylor series, Maclaurin series, polynomial approximation, infinite series, convergence tests