Difficulty: Advanced | Prerequisites: All prior notes (Docs 1, 2, and 3). Strong comfort with integration techniques is essential for convergence tests and Taylor series.
Sequences and series are the defining BC-only material. This is where Calculus BC separates from AB: you need to determine whether an infinite sum converges or diverges, approximate functions with polynomials (Taylor and Maclaurin series), and work with polar coordinates. These topics carry heavy weight on the exam, especially the convergence tests and Taylor series. If you are behind, prioritise the Ratio Test, the Alternating Series Test, and the common Maclaurin series for e^x, sin(x), cos(x), and 1/(1-x).
A sequence is a list of numbers; a series is their sum. Whether that sum converges (lands on a finite value) or diverges (blows up or oscillates) is tested by a battery of convergence tests, each suited to different shapes of series. Taylor series let you represent smooth functions as infinite polynomials, which is useful for approximation. Polar curves use r(theta) instead of y(x), and area inside a polar curve requires a specific integral formula.
Sequence
An ordered list of numbers {a_n}. Think of it as a function whose input is the natural numbers.
Series
The sum of a sequence: sum of a_n. A series converges if the partial sums approach a finite limit.
Geometric series
A series of the form A + Ar + Ar^2 + ... It converges to A/(1-r) when |r| < 1 and diverges otherwise.
Convergence
A series converges if its partial sums approach a finite value. In simple terms, the sum settles down.
Divergence
A series diverges if its partial sums do not approach a finite value.
Taylor series
An infinite polynomial representation of a function centred at x = c, built from the function's derivatives at c. The nth term is f^(n)(c) / n! * (x - c)^n.
Maclaurin series
A Taylor series centred at c = 0. The most common ones to memorise: e^x, sin(x), cos(x), 1/(1-x).
Radius of convergence
The value R such that a power series converges for |x - c| < R and diverges for |x - c| > R. Found using the Ratio or Root Test.
Polar coordinates
A system where each point is described by a distance r from the origin and an angle theta from the positive x-axis. Conversion: x = r cos(theta), y = r sin(theta).
A sequence {a_n} is simply a list of numbers in order. The main question is whether the sequence converges (a_n approaches a limit as n goes to infinity) or diverges.
A series sum of a_n is the sum of all terms of a sequence. You care about whether the infinite sum has a finite value.
The series A + Ar + Ar^2 + Ar^3 + ... converges to A / (1 - r) when |r| < 1. If |r| >= 1, it diverges.
This is the most important specific series to know. Many problems reduce to recognising a geometric series in disguise.
If lim (n->inf) a_n ≠ 0, then the series diverges. This is the first test to try. Note: if the limit is 0, the test is inconclusive (the series might still diverge, e.g. the harmonic series).
If f(x) is positive and decreasing on [k, inf) and f(n) = a_n, then the series and the improper integral from k to infinity converge or diverge together.
Given 0 <= a_n <= b_n for all n:
If sum of b_n converges, then sum of a_n converges.
If sum of a_n diverges, then sum of b_n diverges.
Compare to a known series (p-series or geometric series).
Compute c = lim (n->inf) a_n / b_n. If c is positive and finite (0 < c < inf), then both series converge or both diverge. More flexible than direct comparison.
For a series with terms a_n = (-1)^n * b_n (where b_n >= 0), the series converges if:
lim (n->inf) b_n = 0, and
{b_n} is eventually decreasing.
Compute L = lim (n->inf) |a_(n+1) / a_n|.
L < 1: absolutely convergent.
L > 1: divergent.
L = 1: inconclusive.
Best for series involving factorials or exponentials.
Compute L = lim (n->inf) |a_n|^(1/n).
L < 1: absolutely convergent.
L > 1: divergent.
L = 1: inconclusive.
Useful when the general term has the form (something)^n.
If f(x) is smooth (infinitely differentiable) at x = c, it can be represented by its Taylor series:
f(x) = f(c) + f'(c)(x - c) + f''(c)/2! * (x - c)^2 + ... + f^(n)(c)/n! * (x - c)^n + ...
A Maclaurin series is a Taylor series centred at c = 0.
e^x = 1 + x + x^2/2! + x^3/3! + ... (converges for all x)
sin(x) = x - x^3/3! + x^5/5! - ... (converges for all x)
cos(x) = 1 - x^2/2! + x^4/4! - ... (converges for all x)
1/(1-x) = 1 + x + x^2 + x^3 + ... (converges for |x| < 1)
ln(1+x) = x - x^2/2 + x^3/3 - ... (converges for -1 < x <= 1)
The exam will ask you to write out the first few terms, find a general term, determine the interval of convergence, or use a known series to build a new one (e.g. substituting -x^2 into the e^x series to get e^(-x^2)).
A point is described by (r, theta) rather than (x, y). Conversion formulas:
x = r cos(theta)
y = r sin(theta)
r^2 = x^2 + y^2
tan(theta) = y/x
A = (1/2) * integral from theta_1 to theta_2 of [r(theta)]^2 d(theta)
To find the area between two polar curves, subtract: A = (1/2) * integral of ([R(theta)]^2 - [r(theta)]^2) d(theta).
Finding the correct bounds (theta_1 and theta_2) is the tricky part. Sketch the curve or set r = 0 to find where the curve passes through the origin.
Students often believe that if lim a_n = 0, the series must converge. This is wrong. The harmonic series (sum of 1/n) diverges despite its terms going to zero.
The Ratio Test is inconclusive when L = 1. Students sometimes mistakenly call this convergent.
Students confuse absolute convergence with conditional convergence. Absolute convergence (sum of |a_n| converges) is stronger and implies convergence. Conditional convergence means the series converges, but the series of absolute values diverges.
In Taylor series, students forget to evaluate the nth derivative at the centre c. Each coefficient uses f^(n)(c), not f^(n)(x).
⚠️ The Ratio Test is the go-to for finding the radius of convergence of a power series. Learn to apply it mechanically.
⚠️ Alternating Series Error Bound: the error in truncating an alternating series is less than or equal to the first omitted term. This is a common free-response question.
⚠️ Taylor polynomial approximation questions ("find the third-degree Taylor polynomial") appear nearly every year. Write out terms, do not guess the pattern.
⚠️ Polar area questions require you to square r(theta) before integrating, and to use (1/2) out front. Both are easy to forget.
True or false: If lim (n->inf) a_n = 0, then sum of a_n converges.
Fill in the blank: A geometric series converges when ______.
True or false: The Maclaurin series for sin(x) contains only odd powers of x.
Fill in the blank: The area inside a polar curve r(theta) is given by ______.
True or false: The Ratio Test gives a conclusive answer when L = 1.
Answers: 1. False (necessary but not sufficient). 2. |r| < 1. 3. True. 4. (1/2) integral of [r(theta)]^2 d(theta). 5. False (inconclusive).
Q: Determine whether the series sum of n / (n^2 + 1) converges or diverges.
A: Limit Comparison with 1/n. Compute lim (n->inf) [n/(n^2+1)] / [1/n] = lim n^2/(n^2+1) = 1. Since 1/n (harmonic series) diverges and the limit is positive and finite, the given series also diverges.
Q: Find the radius of convergence of sum of (x^n) / (n * 3^n).
A: Ratio Test. |a_(n+1)/a_n| = |x|/3 * n/(n+1). As n->inf, this approaches |x|/3. Converges when |x|/3 < 1, so |x| < 3. Radius of convergence R = 3.
Q: Write the first four nonzero terms of the Maclaurin series for e^(-x^2).
A: Start with e^u = 1 + u + u^2/2! + u^3/3! + ... Substitute u = -x^2: 1 + (-x^2) + (-x^2)^2/2! + (-x^2)^3/3! = 1 - x^2 + x^4/2 - x^6/6.
Q: Find the area enclosed by the polar curve r = 2cos(theta).
A: The curve traces a complete circle as theta goes from 0 to pi. A = (1/2) integral from 0 to pi of [2cos(theta)]^2 d(theta) = (1/2) integral of 4cos^2(theta) d(theta) = 2 integral of (1 + cos(2theta))/2 d(theta) = integral from 0 to pi of (1 + cos(2theta)) d(theta) = [theta + sin(2theta)/2] from 0 to pi = pi.
Q: Does sum of (-1)^n / sqrt(n) converge? If so, absolutely or conditionally?
A: Alternating Series Test: b_n = 1/sqrt(n) is decreasing and approaches 0. The series converges. But sum of 1/sqrt(n) is a p-series with p = 1/2 < 1, which diverges. So the convergence is conditional.
The convergence tests rely heavily on integration (the Integral Test), limits (the Ratio and Root Tests), and comparison to known series, all skills from the earlier notes.
Taylor series connect calculus to approximation theory. Whenever the exam asks you to estimate a value (like e^0.1 or sin(0.5)), a Taylor polynomial centred at a convenient point is the standard approach.
Polar curves connect to parametric equations, which share similar area and arc length formulas. If you study parametric equations separately, you will notice the polar area formula is a special case.
sequence, series, convergence, divergence, geometric series, p-series, harmonic series, Divergence Test, nth-term test, Integral Test, Comparison Test, Direct Comparison, Limit Comparison Test, Alternating Series Test, Ratio Test, Root Test, absolute convergence, conditional convergence, Taylor series, Taylor polynomial, Maclaurin series, radius of convergence, interval of convergence, power series, polar coordinates, polar curve, polar area, r equals, x equals r cos theta, y equals r sin theta, AP Calculus BC, BC-only topics