The Divergence Test

The divergence test is simply a quick check: if the numbers you're adding don't shrink to zero, the total can't settle on a value. The name tells you what it's able to do: it can only ever prove that a series diverges (never settles), never that it converges. That makes it good as a five-second first look at any series, and it ends a lot of problems on the spot. The statement $\lim_{n\to\infty} a_n \ne 0 \implies \sum a_n$ diverges looks technical, but it's just two pieces: $\lim a_n \ne 0$ (the individual chunks aren't fading to zero) and "diverges" (the running total never lands on a number). It's the most common-sense test in the unit, so don't let the notation scare you.

Why it works. Picture the running total settling on some number $s$. Once it's nearly there, each new chunk can only nudge it a tiny bit, or it would knock the total away from $s$. To say it exactly: you saw last lesson that each term is the jump between consecutive partial sums. If $\sum a_n = s$, then $a_n = s_n - s_{n-1} \to s - s = 0$. So a convergent series always has terms going to 0. Turned around: terms that don't go to 0 mean divergence.

A common mistake. It's tempting to think the reverse holds too: terms go to 0, so the series converges. It doesn't. $\lim a_n = 0$ tells you nothing on its own. The terms can shrink, but too slowly, and still pile up to infinity. The harmonic series is the classic example. Group its terms:

Each bracket is at least $\frac12$ (the group from $\frac{1}{2^{k}+1}$ to $\frac{1}{2^{k+1}}$ has $2^k$ terms, each at least $\frac{1}{2^{k+1}}$). You get infinitely many brackets, each worth at least a half, and infinitely many halves add up to infinity. It grows painfully slowly, though: it takes 12,367 terms to pass 10. So you'd never guess it diverges by adding terms on a computer. You need an argument like this one.

Facts that come with it.

• Changing, adding or removing finitely many terms never changes whether a series converges, only its sum. Convergence is decided by the tail, way out at infinity.
• If $\sum a_n$ converges and $\sum b_n$ diverges, then $\sum (a_n + b_n)$ diverges.

In signal processing and control, terms that refuse to die out mean an unstable system: an echo that never fades, or energy that keeps building up.