Taylor series, sequences and infinite series
From local polynomial approximation and error bounds to sequences, convergence tests and power-series calculus.
This chapter approximates a function near a point by a polynomial built from its derivatives, bounds the error and then asks when the infinite version of that polynomial equals the function. The finite part needs only differentiation and integration by parts; the infinite part needs limits of sequences and tests for the convergence of series, which the middle lessons develop.
Lessons
- Taylor polynomials: finite local models built from derivative data at one centre.
- Taylor's theorem: the exact remainder, error bounds and the classification of stationary points.
- Sequences and limits: what it means for a sequence to converge and how to prove that it does.
- Infinite series: infinite sums defined through their partial sums, with geometric, telescoping and harmonic examples.
- Convergence tests: comparison, integral, ratio, root and alternating tests for series with no closed sum.
- Taylor series: the infinite extension of a Taylor polynomial and the remainder argument that decides when it equals the function.
- Power series: series defined by their coefficients, their radius of convergence and termwise calculus inside that radius.
- Seen questions: past exam questions on Taylor coefficients, remainder bounds, sequence limits, convergence tests and power-series endpoints.
Where it leads
The next chapter carries the finite approximation into several variables: the tangent plane is the first-order Taylor model and the Hessian supplies the quadratic part used to classify stationary points.
Vocabulary checkpoints
- Sequence and series: a sequence is a list of terms; an infinite series is defined by the limit of its partial sums.
- Taylor polynomial and Taylor series: a Taylor polynomial is finite, while a Taylor series represents its function only where the remainder tends to zero.
- Radius and interval of convergence: the radius of a power series fixes the open interval, but each endpoint must be tested separately.
- Pointwise and uniform convergence: pointwise convergence alone does not justify termwise differentiation; power series have the uniform control that does (power series).