`(1)/(4)+(1)/(3)((1)/(4))^(3)+(1)/(5)((1)/(4))^(5)+…..oo=…….`

Options

A. 4 log3
B. log3
C. `(1)/(2)`log5/3
D. `(1)/(2)`log3/5

Answer

To solve the infinite series

\[ S = \frac{1}{4} + \frac{1}{3} \left(\frac{1}{4}\right)^3 + \frac{1}{5} \left(\frac{1}{4}\right)^5 + \ldots \]

we can recognize that this series can be expressed in a more general form.

### Step 1: Rewrite the Series
The series can be rewritten as:

\[ S = \sum_{n=1}^{\infty} \frac{1}{n+2} \left(\frac{1}{4}\right)^{2n-1} \]

This is because the first term corresponds to \(n=1\), the second term corresponds to \(n=2\), and so on.

### Step 2: Identify the Series Type
This series resembles the Taylor series expansion of the logarithmic function. We know that:

\[ \frac{1}{2} \log \left(\frac{1+x}{1-x}\right) = x + \frac{x^3}{3} + \frac{x^5}{5} + \ldots \]

for \(|x| < 1\).

### Step 3: Substitute \(x\)
In our case, we can set \(x = \frac{1}{4}\). Thus, we can write:

\[ \frac{1}{2} \log \left(\frac{1+\frac{1}{4}}{1-\frac{1}{4}}\right) = \frac{1}{2} \log \left(\frac{\frac{5}{4}}{\frac{3}{4}}\right) \]

### Step 4: Simplify the Logarithm
Now simplify the logarithm:

\[ \frac{1}{2} \log \left(\frac{5}{3}\right) \]

### Step 5: Conclusion
Thus, the value of the infinite series \(S\) is:

\[ S = \frac{1}{2} \log \left(\frac{5}{3}\right) \]

### Final Answer
The final answer is:

\[ S = \frac{1}{2} \log \left(\frac{5}{3}\right) \]

---