Statement-I : `(1)/(1.2)+(1)/(2.2^(2))+(1)/(3.2^(3))+….oo=log_(e )1//2`
Statement-II : `((1)/(5)+(1)/(7))+(1)/(3)((1)/(5^(3))+(1)/(7^(3)))+(1)/(5)((1)/(5^(5))+(1)/(7^(5)))+….+oo=(1)/(2)log2`
Which of the above is true

Options

A. Only I
B. Only II
C. both I & II
D. neither I nor II

Answer

To solve the given statements, we will analyze each statement separately and verify their correctness.

### Statement I:
The series given is:
\[ S_1 = \frac{1}{1 \cdot 2} + \frac{1}{2 \cdot 2^2} + \frac{1}{3 \cdot 2^3} + \ldots \]
We can express this series in a more general form:
\[ S_1 = \sum_{n=1}^{\infty} \frac{1}{n \cdot 2^n} \]

To evaluate this series, we can use the known result:
\[ \sum_{n=1}^{\infty} \frac{x^n}{n} = -\ln(1-x) \quad \text{for } |x| < 1 \]
Differentiating both sides with respect to \(x\):
\[ \sum_{n=1}^{\infty} n x^{n-1} = \frac{1}{1-x} \]
Multiplying by \(x\):
\[ \sum_{n=1}^{\infty} n x^n = \frac{x}{(1-x)^2} \]
Now, we can express our series:
\[ \sum_{n=1}^{\infty} \frac{x^n}{n} = -\ln(1-x) \implies \sum_{n=1}^{\infty} \frac{x^n}{n \cdot 2^n} = -\frac{1}{2} \ln(1 - \frac{x}{2}) \]
Setting \(x = 1\):
\[ S_1 = -\frac{1}{2} \ln(1 - \frac{1}{2}) = -\frac{1}{2} \ln(\frac{1}{2}) = \frac{1}{2} \ln(2) \]
Thus, Statement I is true:
\[ S_1 = \frac{1}{2} \ln(2) \]

### Statement II:
The series given is:
\[ S_2 = \left(\frac{1}{5} + \frac{1}{7}\right) + \frac{1}{3}\left(\frac{1}{5^3} + \frac{1}{7^3}\right) + \frac{1}{5}\left(\frac{1}{5^5} + \frac{1}{7^5}\right) + \ldots \]
We can separate the series into two parts:
\[ S_2 = \sum_{n=0}^{\infty} \frac{1}{(2n+1)5^{2n+1}} + \sum_{n=0}^{\infty} \frac{1}{(2n+1)7^{2n+1}} \]

Using the known series expansion:
\[ \sum_{n=0}^{\infty} \frac{x^n}{n} = -\ln(1-x) \]
We can rewrite:
\[ S_2 = \frac{1}{2} \left( \ln\left(\frac{1 + \frac{1}{5}}{1 - \frac{1}{5}}\right) + \ln\left(\frac{1 + \frac{1}{7}}{1 - \frac{1}{7}}\right) \right) \]
Calculating each term:
\[ \frac{1 + \frac{1}{5}}{1 - \frac{1}{5}} = \frac{6}{4} = \frac{3}{2}, \quad \frac{1 + \frac{1}{7}}{1 - \frac{1}{7}} = \frac{8}{6} = \frac{4}{3} \]
Thus:
\[ S_2 = \frac{1}{2} \left( \ln\left(\frac{3}{2}\right) + \ln\left(\frac{4}{3}\right) \right) = \frac{1}{2} \ln(2) \]
Thus, Statement II is also true:
\[ S_2 = \frac{1}{2} \ln(2) \]

### Conclusion:
Both statements are true.