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 first need to identify the type of series we are dealing with.

### Step 1: Identify the series type
The series can be expressed as:
- The first term \( a = 1 \)
- The second term \( 0.6 = \frac{6}{10} = \frac{6}{10^1} \)
- The third term \( 0.06 = \frac{6}{100} = \frac{6}{10^2} \)
- The fourth term \( 0.006 = \frac{6}{1000} = \frac{6}{10^3} \)
- The fifth term \( 0.0006 = \frac{6}{10000} = \frac{6}{10^4} \)

This indicates that the series can be rewritten as:
\[ S = 1 + \frac{6}{10^1} + \frac{6}{10^2} + \frac{6}{10^3} + \ldots \]

### Step 2: Separate the first term
We can separate the first term from the rest of the series:
\[ S = 1 + \left( \frac{6}{10} + \frac{6}{10^2} + \frac{6}{10^3} + \ldots \right) \]

### Step 3: Identify the remaining series
The remaining series \( \frac{6}{10} + \frac{6}{10^2} + \frac{6}{10^3} + \ldots \) is a geometric series where:
- The first term \( a' = \frac{6}{10} \)
- The common ratio \( r = \frac{1}{10} \)

### Step 4: Use the formula for the sum of a geometric series
The sum \( S' \) of an infinite geometric series can be calculated using the formula:
\[ S' = \frac{a'}{1 - r} \]
Substituting the values:
\[ S' = \frac{\frac{6}{10}}{1 - \frac{1}{10}} = \frac{\frac{6}{10}}{\frac{9}{10}} = \frac{6}{9} = \frac{2}{3} \]

### Step 5: Combine the sums
Now, we can combine this with the first term:
\[ S = 1 + S' = 1 + \frac{2}{3} \]

### Step 6: Simplify the final sum
To add \( 1 \) and \( \frac{2}{3} \), we convert \( 1 \) into a fraction:
\[ 1 = \frac{3}{3} \]
Thus,
\[ S = \frac{3}{3} + \frac{2}{3} = \frac{5}{3} \]

### Final Answer
The sum of the series \( S = 1 + 0.6 + 0.06 + 0.006 + 0.0006 + \ldots \) is:
\[ \boxed{\frac{5}{3}} \]