What convergence means for infinite series

For a series ${\sum }_{n=1}^{\infty }{a}_n$, the question is not whether the individual terms 'look small.' The question is whether the partial sums $S_N=a_1+a_2+\cdots +a_N$ approach a finite limit as $N$ grows.

That is why series convergence is a second-layer notion. First you build the sequence of partial sums, then you ask whether that sequence converges.

The geometric series ${\sum }_{n=1}^{\infty }\frac{1}{2^n}$ has partial sums that settle toward $1$. Each new term still changes the total, but by smaller and smaller amounts.

The harmonic series ${\sum }_{n=1}^{\infty }\frac{1}{n}$ behaves differently. Its terms also go to $0$, but the partial sums keep growing. Slowly is not the same as converging.

The spreadsheet makes that contrast concrete: after six terms, the geometric partial sum is already $63/64=0.984375$, while the harmonic partial sum has climbed to about $2.45$ and has no finite stopping point in view.

A necessary condition for convergence is that $a_n\to 0$, but that condition alone does not guarantee that the series converges.

The harmonic series is the standard warning example. Its terms shrink, but the total keeps drifting upward without settling at a finite value.

If sequence convergence is about whether the terms $a_n$ settle, series convergence is about whether the accumulated totals $S_N$ settle.

So the earlier sequence language still applies, but now it is applied to the sequence of partial sums rather than to the original list of terms.