The sum of the series `1+(1)/(4)+(1)/(16)+(1)/(64)+....oo` is

Options

A. Statement-I is true, Statement-II is true, Statement-II is correct explanation for statement-I
B. Statement-I is true, Statement-II is true, Statement-II is NOT a correct explanation for statement-I
C. Statement-I is true, Statement-II is false.
D. Statement-I is false and Statement-II is true.

Answer

To find the sum of the series \( S = 1 + \frac{1}{4} + \frac{1}{16} + \frac{1}{64} + \ldots \), we can recognize that this is a geometric series.

### Step 1: Identify the first term and the common ratio
The first term \( a \) of the series is:
\[ a = 1 \]

The common ratio \( r \) can be found by taking the ratio of the second term to the first term:
\[ r = \frac{\frac{1}{4}}{1} = \frac{1}{4} \]

### Step 2: 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:
\[ S = \frac{a}{1 - r} \]
provided that \( |r| < 1 \).

### Step 3: Substitute the values into the formula
Substituting the values of \( a \) and \( r \) into the formula:
\[ S = \frac{1}{1 - \frac{1}{4}} \]

### Step 4: Simplify the expression
First, simplify the denominator:
\[ 1 - \frac{1}{4} = \frac{4}{4} - \frac{1}{4} = \frac{3}{4} \]
Now substitute this back into the sum formula:
\[ S = \frac{1}{\frac{3}{4}} = 1 \times \frac{4}{3} = \frac{4}{3} \]

### Conclusion
Thus, the sum of the series is:
\[ S = \frac{4}{3} \]

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