Sum to Infinity of a Geometric Progression — Mathematics

Introduction

When the terms of a geometric progression (GP) get progressively smaller in magnitude, the sum of the series approaches a fixed, finite value — even as the number of terms grows without bound. This limiting value is called the sum to infinity. In the 9709 exam, questions on this topic appear regularly, either as standalone calculations or embedded within problems involving algebraic expressions for the first term and common ratio. Mastery requires knowing precisely when a GP converges and how to apply the sum to infinity formula efficiently.

Core Concept

Recall that for a GP with first term $a$ and common ratio $r$ , the sum of the first $n$ terms is:

S_n = \frac{a(1 - r^n)}{1 - r}, \quad r \neq 1

Now consider what happens as $n \to \infty$ . The behaviour of $S_n$ depends entirely on $r^n$ :

If $|r| < 1$ , then $r^n \to 0$ as $n \to \infty$ , so $S_n$ approaches a finite limit.
If $|r| \geq 1$ , then $|r^n|$ does not tend to zero, so $S_n$ diverges (no finite sum to infinity exists).

Convergence condition: A geometric progression converges (has a sum to infinity) if and only if $|r| < 1$ , i.e. $-1 < r < 1$ .

When $|r| < 1$ , letting $n \to \infty$ causes the term $r^n \to 0$ , giving:

S_\infty = \frac{a}{1 - r}

This is the sum to infinity of a convergent GP.

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Introduction

Core Concept

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