It comes last because it needs everything: limits (of sequences now), integration (the integral test), improper integrals (the p-threshold), and derivatives of every order (Taylor).

Warn her about the three weeks. The convergence-test section feels like an unrelated course — a zoo of arbitrary rules about infinite sums with no visible purpose. Then Taylor series arrives and retroactively justifies all of it: the tests were the tools needed to know when an infinite polynomial is legitimate. Knowing the payoff is coming makes the zoo tolerable.

It's also the hardest unit in BC and the largest slice of the exam. Budget accordingly.

An infinite sum is not a sum. You cannot perform infinitely many additions. What you can do is perform each finite prefix and ask where those results are heading.

Zeno's paradoxes are precisely this confusion. To cross a room you must first cross half, then half the remainder, and so on — infinitely many steps, so (Zeno concluded) motion is impossible. The resolution is that ½ + ¼ + ⅛ + ⋯ has partial sums ½, ¾, ⅞, … converging to 1. Infinitely many terms, finite total. That took 2,400 years to state properly, and the reason is that nobody had the concept of a limit.

The tests look like an arbitrary collection. They're all asking one thing: does the tail shrink fast enough? Each test is a different way of measuring "fast enough," and each is really the Unit 1 growth hierarchy in disguise.

A conditionally convergent series can be rearranged to sum to any real number you choose. Any number at all. Or to +∞, or to −∞.

The mechanism: the positive terms alone diverge and the negative terms alone diverge. So take positive terms until you exceed your target, then negatives until you fall below, then positives again — you can steer the partial sums anywhere.

For conditionally convergent series, addition stops being commutative. That's a genuinely disturbing fact about infinity, and it's the entire reason the absolute/conditional distinction exists. Absolutely convergent series are immune — you can reorder them freely.

Look at the first few terms:

• Degree 0: f(a) — a constant approximation.
• Degree 1: f(a) + f′(a)(x−a) — that's linear approximation from Unit 4, verbatim.
• Degree 2: adds f″(a)(x−a)²/2 — the quadratic whose opening direction the second derivative test checks.
• Keep going and you get the whole function.

Every approximation idea in the course was a truncated Taylor series. Linear approximation, differentials, the second derivative test, Newton's method, Euler's method, L'Hôpital — all of them are this, cut off early. That's why the Unit 2 framing "the derivative is the multiplier in the best linear approximation" was worth carrying since September: this unit is where it pays.

Sine is odd, so only odd powers survive; cosine is even, so only even ones. Their coefficients alternate because their derivatives cycle with period 4 (Unit 3 flagged this). e^x has no alternation because every derivative is itself.

Everything else in the course is obtained from these by substitution, differentiation, integration, or multiplication — and doing it that way is far faster and far less error-prone than computing derivatives one at a time.

Students skip the error bound because it's fiddly. It is the part that makes the whole enterprise useful rather than decorative — without it, a truncated series is a guess, and with it, it's a guarantee.

Read the formula: it's the next term you didn't include, with the derivative replaced by its worst-case value on the interval. Nothing more.

The alternating series bound is even simpler and worth preferring when it applies: for an alternating series with decreasing terms, the error is smaller than the first omitted term. One term, no maximization.

This is the largest unit on the exam and reliably the hardest. What follows is the additional depth that the earlier pass compressed: a proper decision procedure, the root test, the tests she'll misapply, and the Taylor manipulations that come up every year.

What is 1 + ¼ + ⅑ + 1/16 + ⋯ ? Posed in 1650, it resisted the Bernoullis for decades — they could prove it converged and could not find the value.

Euler solved it in 1735 at age 28 and got π²/6. Where does π come from in a sum of reciprocal squares? His method — factoring sin x as an infinite product from its roots, treating a transcendental function as a giant polynomial — was completely unjustified by the standards of the time and completely correct. It made him famous across Europe overnight.

1 − ½ + ⅓ − ¼ + ⋯ = ln 2. The same terms, without the signs, diverge to infinity. And by Riemann's theorem, those same terms rearranged can be made to sum to 7, or to π, or to anything you name.

Write out the Maclaurin series for e^x and substitute ix. The powers of i cycle 1, i, −1, −i, with period 4 — the same period as the derivative cycle of sine and cosine, which is not a coincidence.

Separate the real and imaginary terms. The real ones are exactly cosine's series. The imaginary ones are exactly sine's:

Set x = π, and since cos π = −1 and sin π = 0:

Five fundamental constants, three operations, no slack. And it isn't mysticism — it's three series sitting next to each other, and she can do the derivation herself in about six lines.

Save this for the last week. It's the best possible ending to the year, and it only works as an ending if the three series are already familiar.

Next stop — Fourier series. Taylor series builds functions out of powers; Fourier builds them out of sines and cosines. It turns out to be the better basis for anything periodic, and it is the mathematical foundation of signal processing, audio and image compression, MRI reconstruction, and the JPEG format. The conceptual leap — that a function can be decomposed into frequencies — is one of the most consequential ideas in applied mathematics.

Then — complex analysis. Taylor series over the complex numbers behave far better than over the reals: a complex-differentiable function is automatically infinitely differentiable and automatically equal to its Taylor series. Facts that are fussy in real calculus become clean. It also explains something Unit 10 leaves mysterious: why 1/(1+x²) has radius of convergence 1 despite the function being perfectly well-behaved everywhere on the real line. The answer is that it blows up at x = ±i, and the radius of convergence is the distance to the nearest complex singularity. That's a genuinely satisfying loose end to be able to tie off.