Infinite Series & Recurring Decimals

When does an infinite GP have a sum?

A finite GP always has a sum. An infinite geometric series $a+ar+ar^2+\cdots$ converges only when $|r|<1$. In that case

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

If $|r|\ge 1$, the terms do not tend to zero (or fail to shrink fast enough), and there is no finite limiting sum. HSC markers expect you to check $|r|<1$ before writing $S_\infty$.

Worked example 1 — Convergence check

Does $\sum_{n=0}^{\infty} 3\left(-\frac23\right)^n$ converge? If so, find its sum.

Here $a=3$ and $r=-\frac23$. Since $\left|-\frac23\right|<1$, the series converges:

$$S_\infty=\frac{3}{1-(-\frac23)}=\frac{3}{\frac53}=\frac95.$$

If the ratio had been $-\frac32$, you would stop: $|r|>1$, no $S_\infty$.

Recurring decimals as geometric series

A pure recurring decimal such as $0.\dot{2}=0.222\ldots$ is the series

$$\frac{2}{10}+\frac{2}{100}+\frac{2}{1000}+\cdots,$$

an infinite GP with first term $\frac{2}{10}$ and ratio $\frac{1}{10}$. Mixed recurring decimals (e.g. $0.1\dot{2}$) split into a terminating place-value part plus a pure recurring tail.

Worked example 2 — Pure and mixed recurring

Convert $0.2$, $0.\dot{2}$, and $0.1\dot{2}$ to fractions.

• Terminating: $0.2=\frac{2}{10}=\frac15$.
• Pure recurring: $0.\dot{2}$ has $a=\frac{2}{10}$, $r=\frac{1}{10}$, so$$S=\frac{2/10}{1-1/10}=\frac29.$$
• Mixed: $0.1\dot{2}=0.1+0.0\dot{2}$. The tail $0.0\dot{2}$ has $a=\frac{2}{100}$, $r=\frac{1}{10}$:$$\frac{2/100}{1-1/10}=\frac{2}{90}=\frac{1}{45}.$$ Adding $\frac{1}{10}=\frac{9}{90}$ gives $\frac{11}{90}$.

Worked example 3 — Another recurring form

Express $0.\overline{4}2$ ideas carefully: identify which digits repeat, write the first repeating block as $a$, and take $r=10^{-k}$ where $k$ is the repeat length. The same $S_\infty$ formula applies; only $a$ and $r$ change.

For a single repeating digit after the point, denominators of $9$, $99$, $999$ appear naturally from $1-r=1-10^{-k}$.

Exam strategy

1. Write the decimal as a series (or as terminating + series).
2. State $a$ and $r$ explicitly.
3. Verify $|r|<1$ (always true for standard decimal expansions with $r=10^{-k}$).
4. Simplify the fraction fully.

Common mistakes

• Writing $S_\infty$ without checking $|r|<1$ on abstract GPs.
• Treating a mixed recurring decimal as if every digit repeats.
• Off-by-one errors in the first term’s place value ($/10$ vs $/100$).
• Leaving unsimplified fractions when a clear reduction exists.

Practice (hints only)

1. Convert $0.\dot{6}$ and $0.2\dot{7}$ to fractions. (GP with $r=\frac1{10}$.)
2. For which $r$ does $2+2r+2r^2+\cdots$ converge, and what is $S_\infty$? (Need $|r|<1$.)

Where next

With infinite sums under control, study telescoping sums, sigma shifts, and induction — techniques that go beyond a single closed GP formula.