Sum of the series `(sqrt(3) - 1) + 2 (2 - sqrt(3)) + 2 (3 sqrt(3) - 5)+…oo` is

Options

A. `sqrt(3) + 1`
B. `sqrt(3) - 1`
C. `2 - sqrt(3)`
D. none of these

Answer

To find the sum of the series \((\sqrt{3} - 1) + 2(2 - \sqrt{3}) + 2(3\sqrt{3} - 5) + \ldots\), we can recognize that the terms can be expressed in a certain pattern. Let's break it down step by step.

### Step 1: Identify the first few terms
The first few terms of the series are:
1. First term: \(\sqrt{3} - 1\)
2. Second term: \(2(2 - \sqrt{3}) = 4 - 2\sqrt{3}\)
3. Third term: \(2(3\sqrt{3} - 5) = 6\sqrt{3} - 10\)

### Step 2: Rewrite the terms
Notice that we can express the terms in a more structured way:
- The first term can be written as \((\sqrt{3} - 1)\)
- The second term can be expressed as \(2(2 - \sqrt{3}) = 2 \cdot 2 - 2\sqrt{3}\)
- The third term can be expressed as \(2(3\sqrt{3} - 5) = 2 \cdot 3\sqrt{3} - 10\)

### Step 3: Recognize the pattern
We can see that the series can be rewritten in terms of powers of \((\sqrt{3} - 1)\):
- First term: \((\sqrt{3} - 1)^1\)
- Second term: \((\sqrt{3} - 1)^2\)
- Third term: \((\sqrt{3} - 1)^3\)

This suggests that the series can be expressed as:
\[ S = (\sqrt{3} - 1) + (\sqrt{3} - 1)^2 + (\sqrt{3} - 1)^3 + \ldots \]

### Step 4: Identify the series type
The series is a geometric series (GP) where:
- First term \(a = \sqrt{3} - 1\)
- Common ratio \(r = \sqrt{3} - 1\)

### Step 5: Sum of the infinite GP
The sum of an infinite geometric series is given by the formula:
\[ S_{\infty} = \frac{a}{1 - r} \]
Substituting the values:
\[ S_{\infty} = \frac{\sqrt{3} - 1}{1 - (\sqrt{3} - 1)} = \frac{\sqrt{3} - 1}{2 - \sqrt{3}} \]

### Step 6: Rationalize the denominator
To simplify \(\frac{\sqrt{3} - 1}{2 - \sqrt{3}}\), we multiply the numerator and denominator by the conjugate of the denominator:
\[ S_{\infty} = \frac{(\sqrt{3} - 1)(2 + \sqrt{3})}{(2 - \sqrt{3})(2 + \sqrt{3})} \]

Calculating the denominator:
\[ (2 - \sqrt{3})(2 + \sqrt{3}) = 4 - 3 = 1 \]

Calculating the numerator:
\[ (\sqrt{3} - 1)(2 + \sqrt{3}) = 2\sqrt{3} + 3 - 2 - \sqrt{3} = \sqrt{3} + 1 \]

### Step 7: Final result
Thus, the sum of the series is:
\[ S_{\infty} = \sqrt{3} + 1 \]

### Conclusion
The sum of the series \((\sqrt{3} - 1) + 2(2 - \sqrt{3}) + 2(3\sqrt{3} - 5) + \ldots\) is \(\sqrt{3} + 1\).

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