The value of `1-1/20+1/20^2-1/20^3+…..` correct to 5 places of decimal is

Options

A. 1.05
B. 0.95238
C. 0.95239
D. 10.5

Answer

To solve the series \( 1 - \frac{1}{20} + \frac{1}{20^2} - \frac{1}{20^3} + \ldots \) and find its value correct to 5 decimal places, we can recognize that this is an infinite geometric series.

### Step-by-Step Solution:

1. **Identify the first term and common ratio**:
The first term \( a = 1 \) and the common ratio \( r = -\frac{1}{20} \).

2. **Use the formula for the sum of an infinite geometric series**:
The sum \( S \) of an infinite geometric series can be calculated using the formula:
\[ S = \frac{a}{1 - r} \]
where \( |r| < 1 \).

3. **Substitute the values into the formula**:
Here, we have:
\[ S = \frac{1}{1 - \left(-\frac{1}{20}\right)} = \frac{1}{1 + \frac{1}{20}} = \frac{1}{\frac{21}{20}} = \frac{20}{21} \]

4. **Calculate the numerical value**:
Now, we need to compute \( \frac{20}{21} \):
\[ \frac{20}{21} \approx 0.95238095238 \]

5. **Round to 5 decimal places**:
The value \( 0.95238095238 \) rounded to 5 decimal places is \( 0.95238 \).

### Final Answer:
Thus, the value of the series \( 1 - \frac{1}{20} + \frac{1}{20^2} - \frac{1}{20^3} + \ldots \) correct to 5 decimal places is \( \boxed{0.95238} \).