Power & Taylor series
The idea
A power series is a polynomial that never ends: a sum of coefficients times powers of (x − a), convergent on an interval around the center a. Taylor series are the bridge from functions to power series — they rebuild e^x, sin x, cos x, ln(1 + x) out of their derivatives at a single point, with the coefficient of (x − a)^n equal to the nth derivative at a divided by n factorial. This is how calculators evaluate transcendental functions and how physicists replace hard models with manageable polynomial ones.
The working mental model: a degree-n Taylor polynomial is the polynomial that agrees with f at the center to the nth derivative — the best polynomial impersonation of f near a. Truncating leaves an error, and the error is controllable: for an alternating series whose terms shrink steadily to zero, the error is smaller than the first omitted term, and Taylor's remainder theorem bounds it in general. That control is the whole point — an approximation without an error bound is a guess.
Watch the misconception that a Taylor series automatically equals its function everywhere. Convergence holds only on the interval of convergence, and accuracy degrades as you leave the center; the geometric series for 1/(1 − x) is perfect on |x| < 1 and meaningless at x = 2.
Worked example
Use the degree-4 Maclaurin polynomial of cos x to approximate cos(0.2), and bound the error of the approximation.
- Recall the Maclaurin expansion of cosine: cos x = 1 − x²/2 + x⁴/24 − x⁶/720 + …, an alternating series for x = 0.2 with steadily shrinking terms.
- Evaluate the kept terms at x = 0.2: the quadratic term is 0.04/2 = 0.02 and the quartic term is 0.0016/24 ≈ 0.0000667.
- Assemble the degree-4 approximation: cos(0.2) ≈ 1 − 0.02 + 0.0000667 = 0.9800667.
- Bound the error with the alternating series estimate: the error is at most the first omitted term, x⁶/720 = 0.000064/720 ≈ 8.9 × 10⁻⁸, so the approximation is trustworthy to six decimal places.
- Interpret the result: a true value near 0.98006658 confirms it — three polynomial terms reproduce cosine to within about 9 × 10⁻⁸, because powers of the small number 0.2 collapse extremely fast.
Answer. cos(0.2) ≈ 0.9800667, with error below 10⁻⁷ by the alternating series bound.
Check your understanding
- Why do higher powers of (x − a) contribute so little when x is close to the center a?
- How does matching derivatives at one point manage to control a polynomial's behavior on a whole interval?
- What happens to the accuracy of this cosine approximation at x = 2, and which fix would you try?
- How would you explain the alternating series error bound to someone using the picture of partial sums hopping back and forth past the limit?
Build the foundations first
Power & Taylor series builds on these concepts. If any feel shaky, start there.