The sum of the series ( 1+ 0.6 + 0.06 + 0.006 + 0.0006 + ......) is

Options

A. `1(2)/(3) `
B. `1(1)/(3) `
C. `2(1)/(3)`
D. `2(2)/(3)`

Answer

To find the sum of the series \( S = 1 + 0.6 + 0.06 + 0.006 + 0.0006 + \ldots \), we can follow these steps:

### Step 1: Rewrite the Series
The series can be rewritten in fractional form:
\[ S = 1 + \frac{6}{10} + \frac{6}{100} + \frac{6}{1000} + \ldots \]

### Step 2: Identify the First Term and Common Ratio
The first term \( a \) of the series is \( 1 \) and the remaining terms can be represented as a geometric series:
\[ S = 1 + 6 \left( \frac{1}{10} + \frac{1}{100} + \frac{1}{1000} + \ldots \right) \]
Here, the first term of the geometric series is \( \frac{1}{10} \) and the common ratio \( r \) is also \( \frac{1}{10} \).

### Step 3: Sum of the Geometric Series
The sum \( S_g \) of an infinite geometric series can be calculated using the formula:
\[ S_g = \frac{a}{1 - r} \]
Where \( a \) is the first term and \( r \) is the common ratio. For our series:
\[ S_g = \frac{\frac{1}{10}}{1 - \frac{1}{10}} = \frac{\frac{1}{10}}{\frac{9}{10}} = \frac{1}{9} \]

### Step 4: Substitute Back into the Original Series
Now substitute \( S_g \) back into the equation for \( S \):
\[ S = 1 + 6 \cdot \frac{1}{9} \]

### Step 5: Simplify the Expression
Calculate \( 6 \cdot \frac{1}{9} \):
\[ 6 \cdot \frac{1}{9} = \frac{6}{9} = \frac{2}{3} \]
Now, add this to 1:
\[ S = 1 + \frac{2}{3} = \frac{3}{3} + \frac{2}{3} = \frac{5}{3} \]

### Step 6: Final Answer
Thus, the sum of the series is:
\[ S = \frac{5}{3} \]

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