Question

How do you find the sum of the infinite geometric series 0.9+0.09+0.009+…?

Answer 1

Hint: This type of problem is based on the concept of geometric series. First, we have to assume 0.9 to be ‘a’. Then to find the common ratio ‘r’ of the given geometric series, we have to divide the first term and the second term of the series. We get \[r=\dfrac{1}{10}\]. And substitute these values in the formula to find the sum of infinite series of a geometric progression, that is, \[S=\dfrac{a}{1-r}\] where S is the sum of the infinite series.

Complete step-by-step answer:
According to the question, we are asked to find the sum of the infinite geometric series 0.9+0.09+0.009+….
We have been given the geometric series is 0.9,0.09,0.009,…. -----(1)

We know that \[a,ar,a{{r}^{2}},a{{r}^{3}},......\] is a geometric series.
Let us now compare the given series with the standard geometric series.
We get a=0.9
Also we have to find the common ratio ‘r’.
To find the common ratio, let us divide the second term of the series by the first term.
That is, r=\[\dfrac{0.09}{0.9}\].
Let us now multiply in both the numerator and denominator by 100.
We get, \[r=\dfrac{0.09}{0.9}\times \dfrac{100}{100}\]
\[\Rightarrow r=\dfrac{0.09\times 100}{0.9\times 100}\]
On further simplification, we get
\[r=\dfrac{9}{90}\]
We know that 90=9x10.
Therefore, \[r=\dfrac{9}{9\times 10}\]
Cancelling out the common term 9 from the above expression, we get
\[r=\dfrac{1}{10}\]
Hence, the common ratio is \[\dfrac{1}{10}\].
Now, we have to find the sum of the infinite geometric series 0.9+0.09+0.009+…
Let us consider S to be the total sum of the given series.
We know that the formula to find the infinite sum of geometric series is \[\dfrac{a}{1-r}\].
\[\Rightarrow S=\dfrac{a}{1-r}\]
We have found that a=0.9 and \[r=\dfrac{1}{10}\].
Substituting these values in S, we get
\[S=\dfrac{0.9}{1-\dfrac{1}{10}}\]
Taking LCM in the denominator, we get
\[S=\dfrac{0.9}{\dfrac{10-1}{10}}\]
\[\Rightarrow S=\dfrac{0.9}{\dfrac{9}{10}}\]
Now let us multiply 10 in both the numerator and denominator.
We get, \[S=\dfrac{10\times 0.9}{10\times \dfrac{9}{10}}\]
On further simplification, we get
\[\Rightarrow S=\dfrac{9}{9}\]
Therefore, S=1.
Hence, the sum of the infinite geometric series 0.9+0.09+0.009+… is 1.

Note: Whenever you get this type of problems, we should always try to make the necessary calculations in the given series to find the value of ‘a’ and ‘r’. We should avoid calculation mistakes based on sign conventions. If we get r>0, we have to use the formula \[S=\dfrac{a}{r-1}\]. Also, we should not use the formula \[S=\dfrac{a\left( {{r}^{n}}-1 \right)}{r-1}\] to find the sum of infinite series. This formula is normally used to find the sum of countable terms in geometric series.