Adding infinitely many things
$\dfrac{1}{2} + \dfrac{1}{4} + \dfrac{1}{8} + \dfrac{1}{16} + \cdots$ — every step adds something positive. What does the total come to?
At $r = 0.50$ the partial sums climb $0.5, 0.75, 0.875, \dots$ and the stat reads $S_{20} = 0.999999$, hugging a dashed line that sits at exactly $1$. At $r = 0.90$ the climb is slower and the line waits higher, at $9$. Past $r = 1$ there is no line at all: at $r = 1.40$ the trace rockets off the top and $S_{20}$ is already $2924.9$. At $r = -0.50$ the sums zigzag — over, under, over — closing in on $-0.333333$.
A series is a sequence in disguise. The partial sums $S_N$ form a sequence, and the sum of the series is defined as $\lim_{N \to \infty} S_N$ — lesson 1's machinery, reused. For a geometric series with first term $a$ and ratio $r$, a telescoping trick gives a closed form:
Everything hangs on $r^N$. When $|r| < 1$ that factor dies, and $S_N \to \frac{a}{1-r}$. The instrument's first term is $a = r$, so its line sits at $\frac{r}{1-r}$ — at $r = \frac{1}{2}$, that is exactly $1$. Not approximately: the limit of $0.5, 0.75, 0.875, \dots$ is $1$, and the sum of a series means that limit. When $|r| \ge 1$, the factor $r^N$ refuses to die and the series diverges.
About this model
What it represents: the first $20$ terms and partial sums of the geometric series, computed exactly. The $S\infty$ line is the closed form $\frac{r}{1-r}$, not an extrapolation from the bars.
What it leaves out: the other infinitely many terms. Near the boundary the window misleads: at $r = 0.90$ the true sum is $9$ but $S_{20}$ has only reached $7.905810$ — a finite plot cannot distinguish slow convergence from divergence. The formula can, which is why the verdict comes from $|r| < 1$, not from the picture.
Source: the closed form, its telescoping proof, and the convergence condition — Geometric series.
At $r = 0.50$ the stat reads $S_{20} = 0.999999$, and no partial sum ever equals $1$. In what sense does the series sum to $1$?
Using the closed form, evaluate $\dfrac{1}{3} + \dfrac{1}{9} + \dfrac{1}{27} + \cdots$
For which ratios does a geometric series settle to a finite sum?
An infinite sum is not endless adding — it is the limit of the partial sums. The geometric series makes that limit computable: $S_N = a\frac{1-r^N}{1-r}$, and $|r| < 1$ is precisely the condition that kills $r^N$.