For `x in R,`
`tanx+1/2tan""(x)/(2)+1/((2^2))tan""(x)/(2^(2))+...+(1)/(2^(n-1))tan""((x)/(2^(n-1)))` is equal to

Options

A. a. `2cot2x-(1)/(2^(n-1))cos""((x)/(2^(n-1)))`
B. b. `(1)/(2^(n-1))cot""((x)/(2^(n-1)))-2cot2x`
C. c. `cot""((x)/(2^(n-1)))-cot2x`
D. d. none of these

Answer

To solve the problem, we need to evaluate the infinite series given by:

\[ S = \tan x + \frac{1}{2} \tan \left( \frac{x}{2} \right) + \frac{1}{2^2} \tan \left( \frac{x}{2^2} \right) + \ldots + \frac{1}{2^{n-1}} \tan \left( \frac{x}{2^{n-1}} \right) \]

This series can be expressed as:

\[ S = \sum_{k=0}^{n-1} \frac{1}{2^k} \tan \left( \frac{x}{2^k} \right) \]

### Step 1: Recognize the pattern in the series
The series involves terms of the form \(\tan\left(\frac{x}{2^k}\right)\) multiplied by \(\frac{1}{2^k}\).

**Hint:** Look for a way to express the tangent function in terms of cotangent to simplify the series.

### Step 2: Use the identity for tangent
Recall the identity:

\[ \tan A = \frac{\sin A}{\cos A} \]

We can rewrite each term in the series using this identity.

### Step 3: Rewrite the series
Using the identity, we can express the series as:

\[ S = \sum_{k=0}^{n-1} \frac{\sin\left(\frac{x}{2^k}\right)}{\cos\left(\frac{x}{2^k}\right) \cdot 2^k} \]

**Hint:** Consider how to combine these terms or look for a telescoping series.

### Step 4: Consider the limit as \(n\) approaches infinity
As \(n\) approaches infinity, we can analyze the behavior of the series. The series converges to a certain value.

**Hint:** Think about the behavior of \(\tan\) and \(\cot\) functions as the angle approaches zero.

### Step 5: Use the formula for the sum of a geometric series
The series can be related to a geometric series. The sum of a geometric series can be expressed as:

\[ \text{Sum} = \frac{a}{1 - r} \]

where \(a\) is the first term and \(r\) is the common ratio.

### Step 6: Final expression
After manipulating the series and applying the limit, we find that:

\[ S = \frac{\tan x}{1 - \frac{1}{2}} = 2 \tan x \]

Thus, the final answer is:

\[ S = 2 \tan x \]

### Conclusion
The value of the series is \(2 \tan x\).