Find the value of
`(1+ (1)/(2))(1+(1)/(4))(1+(1)/(16))(1+(1)/(156))…oo`.

Options

A. `1`
B. `2`
C. `(1)/(3)`
D. `(1)/(4)`

Explanation

To solve the problem, we need to find the value of the infinite product:

\[ (1 + \frac{1}{2})(1 + \frac{1}{4})(1 + \frac{1}{16}) \ldots \]

Let's denote this infinite product as \( x \).

### Step 1: Define the Infinite Product
We start by defining the product:

\[ x = (1 + \frac{1}{2})(1 + \frac{1}{4})(1 + \frac{1}{16}) \ldots \]

### Step 2: Rewrite the Terms
We can rewrite each term in the product:

\[ x = (1 + \frac{1}{2})(1 + \frac{1}{4})(1 + \frac{1}{16}) = \left(1 + \frac{1}{2}\right) \left(1 + \frac{1}{2^2}\right) \left(1 + \frac{1}{2^4}\right) \ldots \]

### Step 3: Factor Out Common Terms
Notice that each term can be expressed as:

\[ 1 + \frac{1}{2^n} = \frac{2^n + 1}{2^n} \]

Thus, we can express \( x \) as:

\[ x = \frac{3}{2} \cdot \frac{5}{4} \cdot \frac{17}{16} \cdots \]

### Step 4: Multiply Both Sides by \( (1 + \frac{1}{2}) \)
Now, we multiply both sides by \( (1 + \frac{1}{2}) \):

\[ (1 + \frac{1}{2})x = (1 + \frac{1}{2}) \cdot (1 + \frac{1}{2})(1 + \frac{1}{4})(1 + \frac{1}{16}) \ldots \]

This gives us:

\[ \frac{3}{2}x = \frac{3}{2} \cdot (1 + \frac{1}{4})(1 + \frac{1}{16}) \ldots \]

### Step 5: Simplify the Expression
Notice that the right-hand side can be simplified further:

\[ \frac{3}{2}x = (1 + \frac{1}{4})(1 + \frac{1}{16}) \ldots \]

### Step 6: Use the Identity
Using the identity \( (a + b)(a - b) = a^2 - b^2 \), we can analyze the limit of the series as \( n \to \infty \). The terms \( \frac{1}{2^n} \) will approach zero.

### Step 7: Solve for \( x \)
From the equation, we can solve for \( x \):

\[ x = \frac{3}{2} \cdot \frac{1}{1 - \frac{1}{2}} = \frac{3}{2} \cdot 2 = 3 \]

### Final Answer
Thus, the value of the infinite product is:

\[ \boxed{3} \]

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