AP® Calculus BC · Unit 10 · Topic 10.14
The Series Worth Memorizing
Four Maclaurin series — for ex, sin(x), cos(x), and 1/(1-x) — are worth knowing cold, because almost every other series in the unit comes from substituting into or combining these four.
These four series are the building blocks. ex = 1 + x + x2/2! + x3/3! + ... has every power of x divided by its factorial. sin(x) and cos(x) use the same factorial denominators but keep only odd or even powers, with alternating signs: sin(x) = x - x3/3! + x5/5! - ..., cos(x) = 1 - x2/2! + x4/4! - .... And 1/(1-x) = 1 + x + x2 + x3 + ... is just the geometric series, no factorials at all. Once you know these, you rarely derive a new series from scratch — you substitute.
Why these four? ex generates sin(x) and cos(x) through Euler's relation, which is why all three share the factorial pattern. sin(x) is the odd-power half of ex's structure with alternating signs, cos(x) is the even-power half. The geometric series 1/(1-x) is different in kind: it has no factorials because it comes from dividing, not differentiating, and it only converges for |x| < 1, unlike the other three which converge everywhere.
The real skill is substitution and combination: replace x with 2x, -x, or x2 inside a known series, or multiply a series by a polynomial like x2, to build the series for a harder function without starting over. This same trick extends to functions like 1/(1+x2) or ln(1+x), which are really 1/(1-x) in disguise with x replaced by something else. That link is what makes 10.14 the hinge between memorizing series and representing arbitrary functions as power series in the rest of the unit.
Key terms
- Maclaurin series
- A Taylor series centered at x = 0, built from a function's derivatives evaluated at 0.
- Geometric series
- The series 1 + x + x2 + x3 + ..., equal to 1/(1-x) only when |x| < 1.
- Substitution
- Replacing the variable in a known series with an expression like 2x or x2 to get a new series.
- Radius of convergence
- The distance from the center within which a power series adds up to a finite value.
Watch it explained
- How it worksHow are the Maclaurin series for ex, sin(x), cos(x), and 1/(1 - x) related to one another?
- Worked exampleFind the Maclaurin series for cos(2x) by substituting into the cosine series.
- Worked exampleFind the first four nonzero terms of the series for x2 times e−x.
- Common mistakeWhy isn't the series for sin(x) just the cosine series with the signs flipped?
- Ties togetherHow does the memorized series for 1 over (1 - x) reappear when representing other functions as power series?
Check yourself
End of topic.