If `f(x)=(1)/(x+1)+(1)/(2(x+1)^(2))+(1)/(3(x+1)^(3))+…(x gt 1)` and f(1), f(2), f(3) are respectively p, q, r then their ascending order is

Options

A. p, q, r
B. r, p, q
C. r, q, p
D. p, r, q

Answer

To solve the problem, we need to evaluate the function \( f(x) \) defined as follows:

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

This series can be recognized as a logarithmic series. We can express it in a more manageable form.

### Step 1: Rewrite the function
We can rewrite the function \( f(x) \) using the series expansion for logarithms. The series can be expressed as:

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

This series is similar to the Taylor series expansion for \( -\log(1 - t) \) evaluated at \( t = \frac{1}{x+1} \):

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

By substituting \( t = \frac{1}{x+1} \), we have:

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

### Step 2: Evaluate \( f(1) \), \( f(2) \), and \( f(3) \)
Now we can evaluate \( f(1) \), \( f(2) \), and \( f(3) \):

1. **Calculate \( f(1) \)**:
\[ f(1) = \log\left(\frac{1+1}{1}\right) = \log(2) \]
Let \( p = f(1) = \log(2) \).

2. **Calculate \( f(2) \)**:
\[ f(2) = \log\left(\frac{2+1}{2}\right) = \log\left(\frac{3}{2}\right) \]
Let \( q = f(2) = \log\left(\frac{3}{2}\right) \).

3. **Calculate \( f(3) \)**:
\[ f(3) = \log\left(\frac{3+1}{3}\right) = \log\left(\frac{4}{3}\right) \]
Let \( r = f(3) = \log\left(\frac{4}{3}\right) \).

### Step 3: Ascending order of \( p \), \( q \), and \( r \)
Now we need to determine the ascending order of \( p \), \( q \), and \( r \):

- We know that \( \log(2) > \log\left(\frac{4}{3}\right) > \log\left(\frac{3}{2}\right) \) because:
- \( \frac{4}{3} > 2 \) implies \( \log\left(\frac{4}{3}\right) > \log(2) \)
- \( \frac{3}{2} < 2 \) implies \( \log\left(\frac{3}{2}\right) < \log(2) \)

Thus, we can conclude:

\[ \log\left(\frac{3}{2}\right) < \log\left(\frac{4}{3}\right) < \log(2) \]

This means:

\[ q < r < p \]

### Final Result
The ascending order of \( f(1) \), \( f(2) \), and \( f(3) \) is:

\[ \text{Ascending order: } q < r < p \quad \text{(i.e., } \log\left(\frac{3}{2}\right) < \log\left(\frac{4}{3}\right) < \log(2)\text{)} \]