HSC Maths Advanced Syllabus

What is the Limiting Sum Formula?

The limiting sum of a geometric sequence, is also known as the sum of an infinite geometric series. This occurs when sum of the sequence increases indefinitely. This only occurs when the absolute value of the common ratio is less than 1, that is: $|r|<1\quad or \quad -1<r<1$

We use the absolute value of $r$, as the common ratio can be a negative number. The purpose of the limiting sum formula, is to determine whether a geometric series with an infinite number of terms, converges to a finite value.

The Limiting Sum Formula

The limiting sum $S_{\infty}$ of a GP with first term $a$ and common ratio $r$, where

$|r|<1\quad or \quad -1<r<1$ is given by:

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

Recurring Decimals and GP’s

We can also use the limiting sum formula when dealing with recurring decimals. Recurring decimals are just GP’s that continue infinitely, to a number so small it is essentially zero. We implement the limiting sum formula to convert recurring decimals into fractions.

For example, the recurring decimal $0.\overline{33}$ can be expanded into:

$$0.\overline{33}=0.3+0.03+0.003+0.0003+...$$

As such, we can see that a GP with a first term of $0.3$ and a common ratio of $0.1$ has been formed. Thus, we can apply the limiting sum formula.

$$0.\overline{33}=\frac{a}{1-r}\\=\frac{0.3}{1-0.1}\\=\frac{0.3}{0.9}\\=\frac{1}{3}$$

Thus, in applying the limiting sum formula, we have successfully converted the recurring decimal $0.\overline{33}$ into a fraction $\frac{1}{3}$