`(3)/(4.8)+(3.5)/(4.8.12)+(3.5.7)/(4.8.12.16)+….=`

Options

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

Answer

To solve the series

\[ S = \frac{3}{4 \cdot 8} + \frac{3 \cdot 5}{4 \cdot 8 \cdot 12} + \frac{3 \cdot 5 \cdot 7}{4 \cdot 8 \cdot 12 \cdot 16} + \ldots \]

we can rewrite the terms in a more manageable form.

### Step 1: Rewrite the series

Notice that the numerators can be expressed in terms of double factorials. The pattern in the numerators is \(3, 3 \cdot 5, 3 \cdot 5 \cdot 7, \ldots\), which can be generalized as:

\[ \frac{3 \cdot (2n + 1)}{4 \cdot 8 \cdot 12 \cdots (4n)} \]

where \(n\) starts from 0. Thus, we can write:

\[ S = \sum_{n=0}^{\infty} \frac{3 \cdot (2n + 1)!!}{(4n)!!} \]

### Step 2: Factor out constants

We can factor out constants from the series:

\[ S = \frac{3}{4} \sum_{n=0}^{\infty} \frac{(2n + 1)!!}{(4n)!!} \]

### Step 3: Recognize the series

The series can be recognized as a form that can be evaluated using the binomial theorem or generating functions. Specifically, we can relate it to the binomial series expansion:

\[ (1 - x)^{-p} = \sum_{n=0}^{\infty} \frac{(p+n-1)!}{n!(p-1)!} x^n \]

### Step 4: Use the binomial theorem

From the binomial theorem, we can relate our series to a known result. We can express our series in terms of a binomial coefficient:

\[ S = \frac{3}{4} \cdot (1 - \frac{1}{4})^{-3/2} \]

### Step 5: Simplify the expression

Now, simplifying the expression:

\[ S = \frac{3}{4} \cdot (1 - \frac{1}{4})^{-3/2} = \frac{3}{4} \cdot ( \frac{3}{4})^{-3/2} \]

Calculating this gives:

\[ S = \frac{3}{4} \cdot \left(\frac{4}{3}\right)^{3/2} = \frac{3}{4} \cdot \frac{8}{3\sqrt{3}} = \frac{2\sqrt{3}}{3} \]

### Final Result

Thus, the final result of the series is:

\[ S = \frac{2\sqrt{3}}{3} - \frac{3}{4} \]