`(sqrt(3)-1)+(1)/(2)(sqrt(3)-1)^(2)+(1)/(3)(sqrt(3)-1)^(3)+….oo`

Options

A. `log(3+sqrt(2))`
B. `log2`
C. 0
D. `-log(2-sqrt(3))`

Answer

To solve the series

\[ S = (\sqrt{3} - 1) + \frac{1}{2}(\sqrt{3} - 1)^2 + \frac{1}{3}(\sqrt{3} - 1)^3 + \ldots \]

we can recognize that this series resembles the Taylor series expansion for \(-\log(1-x)\):

\[ -\log(1-x) = x + \frac{x^2}{2} + \frac{x^3}{3} + \ldots \]

### Step 1: Identify the variable \(x\)

We can set \(x = \sqrt{3} - 1\). Thus, we can rewrite the series \(S\) as:

\[ S = -\log(1 - (\sqrt{3} - 1)) \]

### Step 2: Simplify the expression inside the logarithm

Now, we simplify \(1 - (\sqrt{3} - 1)\):

\[ 1 - (\sqrt{3} - 1) = 1 - \sqrt{3} + 1 = 2 - \sqrt{3} \]

### Step 3: Substitute back into the logarithm

Now we can substitute this back into our expression for \(S\):

\[ S = -\log(2 - \sqrt{3}) \]

### Step 4: Final expression

Thus, the final expression for the series is:

\[ S = -\log(2 - \sqrt{3}) \]

### Step 5: Verify the answer

To ensure that this is correct, we can check the value of \(2 - \sqrt{3}\) and confirm that it is positive, which it is since \(\sqrt{3} \approx 1.732\) and \(2 - \sqrt{3} \approx 0.268\).

### Conclusion

Therefore, the value of the series is:

\[ S = -\log(2 - \sqrt{3}) \]