Numerical series

$\sum_{k=1}^\infty a_k$ is the limit of the partial sums $S_n = a_1 + \dots + a_n$. Terms going to zero is necessary but not sufficient: the harmonic series $\sum 1/k$ diverges, while $\sum 1/k^2 = \pi^2/6$.

Zeno's paradox (infinitely many steps, finite distance), the decimal expansion of $\pi$, the total cost of an infinite process: we need to know when adding infinitely many numbers gives a finite answer, and how fast the partial sums get there.

The harmonic series is the classic warning. Its terms shrink to zero, but group them: $\frac13 + \frac14 > \frac12$, $\frac15 + \dots + \frac18 > \frac12$, … infinitely many halves. It grows like $\ln n$ — so slowly that the sum of the first $10^{43}$ terms is only about 100, yet it never stops.

$\sum a_k$ converges to $S$ if $S_n = \sum_{k=1}^n a_k \to S$. Necessary condition: $a_k \to 0$. Cauchy criterion: $\sum a_k$ converges iff $\forall\varepsilon\ \exists N:\ |a_{m+1} + \dots + a_n| < \varepsilon$ for $n > m \ge N$.

Algorithm analysis is full of series: the expected number of comparisons of quicksort is $\approx 2n\ln n$ because of $H_n$; the coupon collector needs $nH_n$ draws; amortized costs are geometric sums. In numerics, series give the values of functions — and summing them in floating point needs care.