Find tHe correct value upto 5 places of decimal of `1-1/20+1/(20^2)-1/(20^3)+...`

Options

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

Answer

To find the value of the series \( S = 1 - \frac{1}{20} + \frac{1}{20^2} - \frac{1}{20^3} + \ldots \), we can recognize that this is an infinite geometric series.

### Step 1: Identify the first term and common ratio
The first term \( a \) of the series is \( 1 \) and the common ratio \( r \) is \( -\frac{1}{20} \).

### Step 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 \( |r| < 1 \).

### Step 3: Substitute the values into the formula
Substituting the values of \( a \) and \( r \):
\[ S = \frac{1}{1 - \left(-\frac{1}{20}\right)} = \frac{1}{1 + \frac{1}{20}} = \frac{1}{\frac{21}{20}} = \frac{20}{21} \]

### Step 4: Convert the fraction to decimal
Now, we need to convert \( \frac{20}{21} \) into a decimal. We can do this by performing the division:
\[ 20 \div 21 \approx 0.95238095238\ldots \]

### Step 5: Round to 5 decimal places
Rounding \( 0.95238095238 \) to 5 decimal places gives us:
\[ 0.95238 \]

### Final Answer
Thus, the correct value of the series up to 5 decimal places is:
\[ \boxed{0.95238} \]