`(x)/(x+1)+(1)/(2)((x)/(x+1))^(2)+(1)/(3)((x)/(x+1))^(3)+....=`

Options

A. `log(1+x)`
B. `log(1-x)`
C. `log x`
D. `log((1)/(x))`

Answer

To solve the series given in the question, we need to analyze the expression:

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

This series can be recognized as a power series in the form of:

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

This series resembles the Taylor series expansion for \(-\log(1 - u)\), which is given by:

\[ -\log(1 - u) = \sum_{n=1}^{\infty} \frac{u^{n}}{n} \]

for \(|u| < 1\).

### Step 1: Identify \(u\)

In our case, we can identify \(u\) as:

\[ u = \frac{x}{x+1} \]

### Step 2: Substitute \(u\) into the logarithmic series

Thus, we can rewrite the series \(S\) as:

\[ S = -\log\left(1 - \frac{x}{x+1}\right) \]

### Step 3: Simplify the logarithmic expression

Now, simplify the expression inside the logarithm:

\[ 1 - \frac{x}{x+1} = \frac{x+1 - x}{x+1} = \frac{1}{x+1} \]

### Step 4: Substitute back into the logarithm

Now substituting this back into our expression for \(S\):

\[ S = -\log\left(\frac{1}{x+1}\right) \]

### Step 5: Simplify the logarithm

Using the property of logarithms, we can simplify this further:

\[ S = \log(x+1) \]

### Conclusion

Thus, the final result for the series is:

\[ S = \log(x+1) \]

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