If `|x| lt 1` , then the coefficient of `x^n` in expansion of `(1+x+x^2 + x^3 +…….)^2` is

Options

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

Answer

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

### Step 1: Recognize the series
The series \( 1 + x + x^2 + x^3 + \ldots \) is a geometric series. We can express it using the formula for the sum of an infinite geometric series:
\[ S = \frac{1}{1 - x} \quad \text{for } |x| < 1 \]

### Step 2: Square the series
Now, we need to square this series:
\[ (1 + x + x^2 + x^3 + \ldots)^2 = \left(\frac{1}{1 - x}\right)^2 \]

### Step 3: Simplify the expression
The expression simplifies to:
\[ \left(\frac{1}{1 - x}\right)^2 = \frac{1}{(1 - x)^2} \]

### Step 4: Use the binomial theorem
Using the binomial theorem, we can expand \( \frac{1}{(1 - x)^2} \):
\[ \frac{1}{(1 - x)^2} = \sum_{k=0}^{\infty} \binom{k + 1}{1} x^k = \sum_{k=0}^{\infty} (k + 1)x^k \]

### Step 5: Identify the coefficient of \( x^n \)
From the expansion, the coefficient of \( x^n \) is \( n + 1 \).

### Conclusion
Thus, the coefficient of \( x^n \) in the expansion of \( (1 + x + x^2 + x^3 + \ldots)^2 \) is:
\[ \boxed{n + 1} \]