The sum of the series `3-1+(1)/(3)-(1)/(9)+` . . . .`oo` Is equal to:
(a)`(20)/(9)`
(b)`(9)/(20)`
(c)`(9)/(4)`
(d)`(4)/(9)`

Options

A. `(20)/(9)`
B. `(9)/(20)`
C. `(9)/(4)`
D. `(4)/(9)`

Answer

To find the sum of the series \(3 - 1 + \frac{1}{3} - \frac{1}{9} + \ldots\), we can follow these steps:

### Step 1: Identify the series
The series can be rewritten as:
\[ S = 3 - 1 + \frac{1}{3} - \frac{1}{9} + \ldots \]

### Step 2: Separate the constant term
We can separate the constant term from the rest of the series:
\[ S = 3 + \left(-1 + \frac{1}{3} - \frac{1}{9} + \ldots\right) \]

### Step 3: Identify the remaining series
The remaining series can be expressed as:
\[ T = -1 + \frac{1}{3} - \frac{1}{9} + \ldots \]
This series is a geometric series where the first term \( a = -1 \) and the common ratio \( r = -\frac{1}{3} \).

### Step 4: Find the sum of the geometric series
The sum of an infinite geometric series is given by the formula:
\[ S = \frac{a}{1 - r} \]
For our series \( T \):
- \( a = -1 \)
- \( r = -\frac{1}{3} \)

Substituting these values into the formula:
\[ T = \frac{-1}{1 - \left(-\frac{1}{3}\right)} = \frac{-1}{1 + \frac{1}{3}} = \frac{-1}{\frac{4}{3}} = -\frac{3}{4} \]

### Step 5: Combine the results
Now, we can substitute \( T \) back into the expression for \( S \):
\[ S = 3 + T = 3 - \frac{3}{4} \]

### Step 6: Simplify the expression
To combine these terms, convert \( 3 \) into a fraction:
\[ S = \frac{12}{4} - \frac{3}{4} = \frac{12 - 3}{4} = \frac{9}{4} \]

### Conclusion
Thus, the sum of the series is:
\[ S = \frac{9}{4} \]

The correct answer is option (c) \( \frac{9}{4} \).

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