0.3,0.18,0.108,………….to `oo`

Answer

To solve the problem of finding the sum of the infinite series 0.3, 0.18, 0.108, ..., we first need to identify the nature of the series.

### Step-by-Step Solution:

1. **Identify the First Term (a)**:
The first term of the series is given as:
\[ a = 0.3 \]

**Hint**: Look for the first number in the sequence to identify 'a'.

2. **Determine the Common Ratio (r)**:
To find the common ratio \( r \), we can divide the second term by the first term:
\[ r = \frac{0.18}{0.3} = 0.6 \]

**Hint**: The common ratio in a geometric series can be found by dividing any term by the previous term.

3. **Check the Condition for Convergence**:
Since \( |r| < 1 \) (in this case, \( 0.6 < 1 \)), the series is convergent, and we can use the formula for the sum of an infinite geometric series.

**Hint**: For a geometric series to converge, the absolute value of the common ratio must be less than 1.

4. **Use the Formula for the Sum of an Infinite Geometric Series**:
The formula for the sum \( S \) of an infinite geometric series is given by:
\[ S = \frac{a}{1 - r} \]
Substituting the values of \( a \) and \( r \):
\[ S = \frac{0.3}{1 - 0.6} = \frac{0.3}{0.4} \]

**Hint**: Remember to substitute the values carefully into the formula.

5. **Calculate the Sum**:
Now, we can simplify the expression:
\[ S = \frac{0.3}{0.4} = 0.75 \]

**Hint**: When dividing decimals, you can convert them to fractions for easier calculations.

### Final Answer:
The sum of the infinite series is:
\[ S = 0.75 \]