If `(1)/(n+1)+(1)/(2(n+1)^(2))+(1)/(3(n+1)^(3))+….=`
`lambda((1)/(n)-(1)/(2n^(2))+(1)/(3n^(3))-……)` then `lambda=`

Options

A. 2
B. `n+1`
C. 1
D. n

Answer

To solve the problem, we need to analyze the infinite series given on the left-hand side and relate it to the series on the right-hand side, which involves the logarithmic functions. Let's break it down step by step.

### Step 1: Identify the Left-Hand Side Series
The left-hand side of the equation is:
\[ S = \frac{1}{n+1} + \frac{1}{2(n+1)^2} + \frac{1}{3(n+1)^3} + \ldots \]
This series can be expressed in terms of the logarithmic function.

### Step 2: Recognize the Series as a Logarithm
We can relate this series to the logarithmic series:
\[ \log(1 + x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \ldots \]
If we let \( x = \frac{1}{n+1} \), then:
\[ \log(1 + \frac{1}{n+1}) = \frac{1}{n+1} - \frac{1}{2(n+1)^2} + \frac{1}{3(n+1)^3} - \ldots \]
Thus, the left-hand side can be written as:
\[ S = -\log\left(1 - \frac{1}{n+1}\right) \]

### Step 3: Simplify the Logarithmic Expression
Now, simplifying the logarithm:
\[ -\log\left(1 - \frac{1}{n+1}\right) = -\log\left(\frac{n}{n+1}\right) = \log\left(\frac{n+1}{n}\right) \]

### Step 4: Analyze the Right-Hand Side Series
The right-hand side of the equation is:
\[ \lambda\left(\frac{1}{n} - \frac{1}{2n^2} + \frac{1}{3n^3} - \ldots\right) \]
This series can also be related to the logarithmic function:
\[ \log(1 + \frac{1}{n}) = \frac{1}{n} - \frac{1}{2n^2} + \frac{1}{3n^3} - \ldots \]

### Step 5: Set the Two Sides Equal
Now we can equate both sides:
\[ \log\left(\frac{n+1}{n}\right) = \lambda \log\left(1 + \frac{1}{n}\right) \]

### Step 6: Simplify the Equation
We know that:
\[ \log\left(\frac{n+1}{n}\right) = \log\left(1 + \frac{1}{n}\right) \]
Thus, we can write:
\[ \log\left(1 + \frac{1}{n}\right) = \lambda \log\left(1 + \frac{1}{n}\right) \]

### Step 7: Solve for \(\lambda\)
Since the logarithmic terms are equal, we can divide both sides by \(\log\left(1 + \frac{1}{n}\right)\) (assuming it is not zero):
\[ 1 = \lambda \]
Thus, we find:
\[ \lambda = 1 \]

### Final Answer
Therefore, the value of \(\lambda\) is:
\[ \lambda = 1 \]