Maths (university) · Part 07
Sequences and series
Adding infinitely many numbers, and when the total settles down.
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- Sequence limitA sequence is a list of numbers a_1, a_2, a_3, and so on. Its limit is the value the terms approach as n gets very large.
- Partial sumThe sum of the first n terms of a series. The infinite series equals the limit of its partial sums, if that limit exists.
- Geometric seriesA series where each term is a fixed multiple r of the one before. It adds to a / (1 - r) when |r| < 1, and diverges otherwise.
- Divergence testIf the terms of a series do not tend to 0, the series diverges. Terms tending to 0 is necessary, but does not guarantee convergence.
- p-seriesThe series of 1 / n^p converges if p > 1 and diverges if p <= 1. With p = 1 it is the harmonic series, which diverges slowly.
- Ratio testFind the limit L of |a_(n+1) / a_n|. If L < 1 the series converges, if L > 1 it diverges, and if L = 1 the test says nothing.
- Alternating seriesA series whose terms switch sign. If the terms shrink steadily towards 0, it converges. The error is smaller than the first term you leave out.
- Radius of convergenceA power series (a sum of terms a_n x^n) converges for x with |x| < R, where R is the radius of convergence. Test the ends separately.
- Taylor seriesA function written as an infinite sum of powers of x, with coefficients f^(n)(0) / n!. Partial sums give polynomial approximations.