Foundations — Sum of infinite GP — when it converges, proof

The parent note proves a formula for the total of an infinite GP. Before you can trust that formula, you must own every symbol in it. This page assumes nothing. We build each idea, attach it to a picture, and only then let the next idea use it.

1. A sequence — numbers in a line

The picture: think of a row of stepping stones, each carrying a number. You stand on stone 1, then stone 2, then stone 3. The index is the stone's position; the value is what's painted on it.

Why the topic needs it: an infinite GP is a sequence. Every term-by-term claim in the parent ("terms shrink," "partial sums settle") is a statement about this list. See Limits of Sequences for what "settle" will eventually mean.

2. First term $a$ — where the list starts

The picture: it's the number painted on stone 1 in the figure above.

Why the topic needs it: every term of a GP is built by starting at $a$ and multiplying. If you grab the wrong starting number, everything after is wrong.

3. Common ratio $r$ — the fixed multiplier

Here is the idea that makes a GP a GP, and not just any sequence.

Why a ratio (division) and not a difference (subtraction)? We use division on purpose. A GP grows by repeated multiplication, so the thing that stays constant is the ratio of neighbours, not their gap. Dividing a term by the one before it cancels the multiplication and leaves exactly the multiplier $r$ behind. (If instead the gap were constant, you'd have an arithmetic progression — a different beast.)

The picture: picture each arrow from one stone to the next labelled "$\times r$." Same label on every arrow — that sameness is what "common" means.

4. Powers and the exponent $n$ — "multiply by $r$, $n$ times"

Once we have $a$ and $r$, every term is $a$ multiplied by $r$ some number of times. We need shorthand for "multiply by $r$, $n$ times."

The picture: starting from $a$, each arrow "$\times r$" you cross adds one to the exponent. After $n$ arrows you have $a{r}^n$.

Why the topic needs it: it lets us write the whole endless list compactly — $a,ar,a{r}^2,a{r}^3,\dots$ and it is the exponent-carrying term $r^N$ that the parent's whole convergence argument hinges on. See Geometric Progression — nth term for the general term.

5. The summation symbol $\sum$ — "add all these up"

We keep writing $a+ar+a{r}^2+\cdots$. Mathematicians pack that into one symbol.

The picture: $\sum$ is a machine with a dial ($n$) that clicks $0,1,2,3,\dots$; at each click it drops the term $a{r}^n$ into a growing pile.

Why the topic needs it: it is the exact object we are trying to give a finite value to. "Does the pile stop growing at a finite height?" is the convergence question. See Convergence of Series.

6. Infinity $\infty$ — the "never stops" symbol

The picture: the sideways loop $\infty$ is a road with no far end — you can always drive further. You never arrive at $\infty$; you head toward it.

Why the topic needs it: an infinite GP has $\infty$ many terms, and the key move is watching what happens as $N\to \infty$. Confusing $\infty$ for a normal number is exactly how people "prove" nonsense like $1+2+4+\cdots =-1$.

7. Partial sum $S_N$ and the infinite sum $S_{\infty }$

We can't literally add infinitely many numbers in one go. So we add a finite chunk and watch it.

The picture: the growing pile-height after you've dropped in $N$ terms. $S_1$ is one block tall, $S_2$ a bit taller, and so on — and $S_{\infty }$ is the ceiling the pile-height creeps up to.

Why the topic needs it: watching $S_N$ is the only honest way to make sense of an endless sum. The parent's whole result is a rule for when $S_{\infty }$ exists and what it equals. See Sum of finite GP for the closed form of $S_N$.

8. Absolute value $|r|$ — size, ignoring the sign

The parent's whole verdict is the condition $|r|<1$. Unpack those bars.

The picture: on a number line, $|r|$ is the distance from $0$ to $r$. Both $+\frac{1}{2}$ and $-\frac{1}{2}$ sit half a step from zero, so both have $|r|=\frac{1}{2}$.

Why the topic needs it: whether terms shrink depends on size, not sign. $r=-\frac{1}{2}$ still shrinks (it just flips sign each time). The clean condition is therefore "$|r|<1$" — meaning $r$ lies strictly between $-1$ and $1$, i.e. closer to $0$ than a full step.

How the foundations feed the topic

Equipment checklist

Cover the right-hand side and test yourself before entering the parent proof.

Connections

• Parent: Sum of infinite GP
• Geometric Progression — nth term
• Sum of finite GP
• Limits of Sequences
• Convergence of Series
• Recurring Decimals as Fractions
• Present Value & Perpetuities