How Do You Find the Sum of an Infinite Convergent Geometric Series?

To find the sum of an infinite convergent geometric series, identify the first term $u_1$ and common ratio $r$, confirm that $\lvert r\rvert<1$, then use

$$S_{\infty}=\frac{u_1}{1-r}.$$

This is HL-only content in Mathematics: Applications and Interpretation, subtopic AHL 1.11. It may be tested on any GDC-active HL paper.

Why the Formula Works

A geometric series has a constant ratio between consecutive terms. Dividing any term by the preceding term gives the common ratio:

$$r=\frac{u_{n+1}}{u_n}.$$

The sum of the first $n$ terms is

$$S_n=\frac{u_1(1-r^n)}{1-r}.$$

When $\lvert r\rvert<1$, successive powers of $r$ approach zero. Therefore,

$$S_{\infty}=\lim_{n\to\infty}\frac{u_1(1-r^n)}{1-r}=\frac{u_1}{1-r}.$$

The condition must be checked before substitution. For example, $5+10+20+\cdots$ has $r=2$. Although it is geometric, $\lvert r\rvert\geq1$, so its partial sums do not approach a finite limit. Applying the sum-to-infinity formula would therefore be invalid.

Worked Examples

For

$$12+6+3+1.5+\cdots,$$

$u_1=12$ and $r=\frac{6}{12}=0.5$. Since ,

$$S_{\infty}=\frac{12}{1-0.5}=24.$$

For

$$8-4+2-1+\cdots,$$

$u_1=8$ and $r=-0.5$. Therefore,

$$S_{\infty}=\frac{8}{1-(-0.5)}=\frac{16}{3}.$$

The terms alternate in sign, but their magnitudes decrease toward zero. The partial sums consequently approach $\frac{16}{3}$.

A common misconception is that $r$ must be positive. This is incorrect: convergence depends on $\lvert r\rvert<1$, not the sign of $r$. Terms approaching zero alone also do not guarantee convergence for every type of series.

In a modelling context, the limiting sum represents the total accumulated amount after indefinitely many stages. For example, if a ball travels 3 m on its first bounce and each later bounce is 60% of the previous distance, the total upward distance is $3/(1-0.6)=7.5$ m. The ratio is $0.6$, so convergence is guaranteed. This interpretation links the calculation to the AI modelling focus.

Exam Technique

State the convergence condition before applying the formula, and show how you calculated $r$. Do not confuse the infinite-series formula with the finite-series formula. Keep full GDC precision and round only the final answer, normally to three significant figures unless an exact value is appropriate.