Sequences and series in calculus
Adding infinitely many numbers can never be finished. When does such a sum still have a value, and how can a function like sine be written as an infinitely long polynomial?
Where you are 8 inside
What this area is about
A series \sum a_n has a sum when its partial sums S_N = a_1 + \cdots + a_N approach a number. Deciding when that happens is the first half of this area. The second half lets the terms carry a variable, so that a series in powers of x - a can stand for a function, and a calculator that only adds and multiplies can compute e^x or \sin x.
How these fit together
Foundations. Convergence of sequences says when a list of numbers settles. Infinite series and partial sums defines a series’ sum as the limit of its partial sums.
Deciding convergence. The nth term test says a series whose terms do not approach 0 diverges. The geometric series test says \sum a r^n with a \ne 0 converges exactly when \lvert r \rvert < 1. Convergence tests handle the series with no formula for their sums.
Functions as series. Power series converge on an interval, whose endpoints need separate checks. Representing functions as power series builds new series from the geometric one by substitution, differentiation and integration. Taylor series builds a series from a function’s derivatives at one point.
A path through it
Sequences, then partial sums. The nth term test before the geometric series test, which uses it; then the convergence tests. Power series need the convergence tests, representing functions needs power series and geometric series, and Taylor series come last.
Later, beyond this course, come Taylor’s theorem with remainder and power series solutions of differential equations.