Infinite Series (Leaving Cert Mathematics): Revision Notes

Infinite Series

Introduction

Let's consider a simple geometric sequence with a starting term of $1$ and a common ratio of $\tfrac{1}{2}$.

$$1, \tfrac{1}{2}, \tfrac{1}{4}, \tfrac{1}{8}, ...$$

Now let's take some geometric series from this sequence :

• $S_1=1$
• $S_2=1+\tfrac{1}{2}=1.5$
• $S_3=1+\tfrac{1}{2}+\tfrac{1}{4}=1.75$
• $S_4=1+\tfrac{1}{2}+\tfrac{1}{4}+\tfrac{1}{8}=1.875$
• $S_5=1+\tfrac{1}{2}+\tfrac{1}{4}+\tfrac{1}{8}+\tfrac{1}{16}=1.9375$
• $S_6=1+\tfrac{1}{2}+\tfrac{1}{4}+\tfrac{1}{8}+\tfrac{1}{16}+\tfrac{1}{32}=1.96875$
• $S_7=1+\tfrac{1}{2}+\tfrac{1}{4}+\tfrac{1}{8}+\tfrac{1}{16}+\tfrac{1}{32}+\tfrac{1}{64}=1.984375$ An observation that we can make is that adding incremental terms to each sum contributes less to the total sum :

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If we take the sum to infinity, then the total sum will converge to some value since adding incremental terms affects the sum minimally as $n$ increases.

This only applies if $|r|<0$.

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

Example

Recurring Decimals

You might have seen decimal notation that feature a dot on top of a number or set of numbers in the decimal. This is known as a repeating decimal, meaning that digit repeats to infinity.

$$\begin{align*} 0.1\dot{2}&=0.12222222222222222222222... \\\\ 4.\dot{2}\dot{3}&=4.23232323232323232323232... \end{align*}$$

The sum to infinity can be used to write these decimal numbers as fractions.

Example

Example