What is the coefficient of `x^(4)` in the expansion of `(1+2x+3x^(2)+4x^(3)+….)^(1//2)` ?

Options

A. `1//4`
B. `1//16`
C. 1
D. `1//128`

Explanation

To find the coefficient of \( x^4 \) in the expansion of \( (1 + 2x + 3x^2 + 4x^3 + \ldots)^{1/2} \), we can follow these steps:

### Step 1: Identify the series
The series \( 1 + 2x + 3x^2 + 4x^3 + \ldots \) can be expressed as:
\[ S = \sum_{n=0}^{\infty} (n+1)x^n \]
This is a power series where the coefficient of \( x^n \) is \( n+1 \).

### Step 2: Find the closed form of the series
The series can be simplified using the formula for the sum of a power series. The sum can be derived as follows:
\[ S = \frac{1}{(1-x)^2} \]
This is obtained by differentiating the geometric series \( \frac{1}{1-x} \).

### Step 3: Substitute into the expression
Now, substituting \( S \) into our expression, we get:
\[ S^{1/2} = \left(\frac{1}{(1-x)^2}\right)^{1/2} = \frac{1}{\sqrt{(1-x)^2}} = \frac{1}{1-x} \]

### Step 4: Expand the expression
Next, we need to expand \( \frac{1}{1-x} \) using the binomial series:
\[ \frac{1}{1-x} = \sum_{n=0}^{\infty} x^n \]
This series gives us the coefficients for each power of \( x \).

### Step 5: Find the coefficient of \( x^4 \)
From the expansion \( \sum_{n=0}^{\infty} x^n \), we see that the coefficient of \( x^4 \) is simply \( 1 \).

### Conclusion
Thus, the coefficient of \( x^4 \) in the expansion of \( (1 + 2x + 3x^2 + 4x^3 + \ldots)^{1/2} \) is \( 1 \).