`(1)/(log_(2) e) + (1)/(log_(2)e^(2)) + (1)/(log_(2) e^(4)) ` + .... =

Options

A. `2 log_(e) ` 2
B. `log_(e)` 3
C. `log_(e)`2
D. none of these

Answer

To solve the expression

\[ \frac{1}{\log_2 e} + \frac{1}{\log_2 e^2} + \frac{1}{\log_2 e^4} + \ldots \]

we can follow these steps:

### Step 1: Rewrite the logarithms
Using the property of logarithms that states \(\log_b a^m = m \cdot \log_b a\), we can rewrite each term in the series:

\[ \frac{1}{\log_2 e^n} = \frac{1}{n \cdot \log_2 e} \]

Thus, the series can be rewritten as:

\[ \frac{1}{\log_2 e} \left( \frac{1}{1} + \frac{1}{2} + \frac{1}{4} + \ldots \right) \]

### Step 2: Identify the series
The series inside the parentheses is:

\[ 1 + \frac{1}{2} + \frac{1}{4} + \ldots \]

This is a geometric series where the first term \(a = 1\) and the common ratio \(r = \frac{1}{2}\).

### Step 3: Sum the geometric series
The sum \(S\) of an infinite geometric series can be calculated using the formula:

\[ S = \frac{a}{1 - r} \]

Substituting the values:

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

### Step 4: Substitute back into the expression
Now, substituting back into our expression, we have:

\[ \frac{1}{\log_2 e} \cdot 2 = \frac{2}{\log_2 e} \]

### Step 5: Use the change of base formula
Using the change of base formula for logarithms, we know:

\[ \log_2 e = \frac{1}{\log_e 2} \]

Thus, we can rewrite:

\[ \frac{2}{\log_2 e} = 2 \cdot \log_e 2 \]

### Final Answer
So, the final value of the original expression is:

\[ 2 \cdot \log_e 2 \]