chapter 03

Infinite sums

Adding without stopping does not always mean growing without bound.

What does it mean to add forever?

An infinite sum is understood through its partial sums. Add the first term, then the first two, then the first three, and watch what those finite totals do.

Consider a geometric series:

$$\frac12+\frac14+\frac18+\frac1{16}+\cdots=1.$$

$1/2$$1/4$$1/8$$1/16$…

The total never reaches 1 after finitely many steps, but it can get arbitrarily close. We say the series converges to 1.

Small terms are not enough

The terms of the harmonic series shrink to zero, yet its partial sums grow without bound:

$$1+\frac12+\frac13+\frac14+\cdots=\infty.$$

Group the terms after 1 into blocks of lengths $1,2,4,8,\ldots$. Every block contributes at least $1/2$. Infinitely many half-units force the total upward forever.

$1$$\frac12$$\frac13+\frac14$$\frac15+\cdots+\frac18$next block

Powers change everything

Replace $1/n$ with $1/n^s$. For real $s>1$, the terms shrink quickly enough for the series to converge. At $s=2$, Euler famously found

$$1+\frac1{2^2}+\frac1{3^2}+\frac1{4^2}+\cdots=\frac{\pi^2}{6}.$$

The exponent $s$ acts like a dial controlling how strongly large integers are suppressed. That dial becomes our function’s input.