Sum of an Infinite Geometric Series
To examine the sum of all the terms of an infinite geometric sequence, we need to consider
when n gets very large.
|r| is the size of r. |r|>1 means r<-1 or r>1.
If |r|>1, the series is said to be divergent and the sum becomes infinitely large.
For example, when r=2, 1+2+4+8+16+… is infinitely large.
If |r|<1, or in other words -1<r<1, then as n becomes very large, rn approaches 0.
For example, when r=½, 1+½+¼+⅛+… =2.
This means that Sn will get closer and closer to u1/(1-r).
If |r|<1, an infinite geometric series of the form u1+u1⋅r+u1⋅r2+⋯=∑∞k=1 u1⋅rk-1
will converge to the sum
We call this the limiting sum of the series.
This result can be used to find the value of recurring decimals.
An application of the sum of infinite geometric series is expressing non-terminating, recurring decimals as rational numbers.
Note: The first five letters in the word ‘rational’ spell ‘ratio’. In other words, a rational number is any number that can be written as a ratio (i.e. a fraction).
When we attempt to express a common fraction such as ⅜ or as 4/11 as a decimal fraction, the decimal always either terminates or ultimately repeats.
Thus 0.375=⅜ (Decimal terminate), 4/11=0.363636… (Decimal repeats)
We can express the recurring decimal fraction 0.36 (or 0.3 ̇6 ̇) as a common fraction.
The bar (0.36) means that the numbers appearing under it are repeated endlessly. i.e. 0.36 means 0.363636… .
Thus a non-terminating decimal fraction in which some digits are repeated again and again in the same order in its decimal parts is called a recurring decimal fraction.
Recurring decimals can be expressed neatly by placing a stripe over the first and last figures which repeat.
This is called the overline notation. For example:
💎 Worked Example 1.
Express the recurring decimal 0.13131313… as a proper fraction.
(1) Express the given number as a geometric series.
(2) State the values of a and r. a=0.13 and r=0.0013/0.13=0.01
(3) find the sum to infinity, S∞
Write the formula for the sum to infinity.
(4) Substitute values of a and r into the formula and simplify.
(5) Multiply both numerator and denominator by 100 to get rid of the decimal point.
📌 Example 1. By writing the recurring decimal 0.12=0.121212… as
express 0.12 as a rational number in simplest form.
We can write the decimal 0.12 as
which is a geometric series with a=12/102 and r=1/102. The limiting sum is
📌 Ex2. find the fraction equivalent to the recurring decimals 0.123.
Here a=0.123, r=0.000123÷0.123=0.001
📌 Ex3. Express the recurring decimal 0.73 in the form a/b, where a, b∈N.
The series in the brackets is an infinite geometric series, with a=0.03 and r=0.003÷0.03=0.1.
💪 Let’s read post •Writing a Repeating Decimal as a Fraction with three methods👈.