Find `1-(1)/(2) + (1)/(4) + (1)/(8) + (1)/(16) - (1)/(32) + .......... oo`

Explanation

To solve the series \( S = 1 - \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} - \frac{1}{32} + \ldots \), we can break it down step by step.

### Step 1: Identify the series components
The series can be rewritten as:
\[ S = 1 + \left(-\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} - \frac{1}{32}\right) \]

### Step 2: Group the terms
We can group the positive and negative terms:
\[ S = 1 + \left(-\frac{1}{2} + \left(\frac{1}{4} + \frac{1}{8} + \frac{1}{16}\right) - \frac{1}{32}\right) \]

### Step 3: Identify the geometric series
The positive terms \( \frac{1}{4} + \frac{1}{8} + \frac{1}{16} \) can be recognized as a geometric series:
- First term \( a = \frac{1}{4} \)
- Common ratio \( r = \frac{1}{2} \)

### Step 4: Sum the geometric series
The sum of an infinite geometric series is given by the formula:
\[ S = \frac{a}{1 - r} \]
For our series:
\[ S = \frac{\frac{1}{4}}{1 - \frac{1}{2}} = \frac{\frac{1}{4}}{\frac{1}{2}} = \frac{1}{4} \cdot 2 = \frac{1}{2} \]

### Step 5: Combine the results
Now, substituting back into our expression for \( S \):
\[ S = 1 - \frac{1}{2} + \frac{1}{2} - \frac{1}{32} \]

### Step 6: Simplify the expression
Now, we can simplify:
\[ S = 1 - \frac{1}{32} \]
\[ S = \frac{32}{32} - \frac{1}{32} = \frac{31}{32} \]

### Final Result
Thus, the sum of the series is:
\[ S = \frac{31}{32} \]

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