Series
Infinite sums and when they make sense: the tests for convergence, and their use in analysing algorithms, discounting rewards and summing numbers safely in floating point.
8 topics
Topics
Numerical series
is the limit of the partial sums . Terms going to zero is necessary but not sufficient: the harmonic series diverges, while .
Geometric series
for . The one series everyone can sum in closed form — and the one behind dynamic arrays, divide-and-conquer recurrences and the discounted rewards of reinforcement learning.
Comparison tests
For positive terms: if and converges, so does . The limit version: if , both series behave the same. In practice, compare with geometric or -series.
Integral test
For positive decreasing , and converge or diverge together, and the integral estimates the sum: that is how is obtained.
Ratio test
If : converges, diverges, says nothing. It compares the series with a geometric one, and it is the natural test for series with factorials.
Root test
If : converges, diverges. Slightly stronger than the ratio test; it gives the radius of convergence of power series (Cauchy–Hadamard).
Alternating series
with decreasing to 0 converges (Leibniz), and the error of stopping at term is at most the first omitted term — a free error bound.
Absolute and conditional convergence
converges absolutely if does. Absolutely convergent series can be reordered freely; conditionally convergent ones cannot — Riemann showed a rearrangement can make them sum to any number.