The coefficient of `x^4` in `((1+x)/(1-x))^2,|x|<1`, is

Options

A. 4
B. -4
C. `10+.^(4)C_(2)`
D. `16`

Answer

To find the coefficient of \( x^4 \) in the expansion of \( \left( \frac{1+x}{1-x} \right)^2 \) for \( |x| < 1 \), we can follow these steps:

### Step 1: Rewrite the expression
We start by rewriting the expression:
\[ \left( \frac{1+x}{1-x} \right)^2 = (1+x)^2 (1-x)^{-2} \]

### Step 2: Expand \( (1+x)^2 \)
Next, we expand \( (1+x)^2 \):
\[ (1+x)^2 = 1 + 2x + x^2 \]

### Step 3: Expand \( (1-x)^{-2} \)
Now, we need to expand \( (1-x)^{-2} \) using the binomial series:
\[ (1-x)^{-2} = \sum_{n=0}^{\infty} \binom{n+1}{1} x^n = \sum_{n=0}^{\infty} (n+1)x^n \]

### Step 4: Combine the expansions
Now we will combine the expansions:
\[ (1 + 2x + x^2) \cdot \sum_{n=0}^{\infty} (n+1)x^n \]

### Step 5: Find the coefficient of \( x^4 \)
To find the coefficient of \( x^4 \), we consider the contributions from each term in \( (1 + 2x + x^2) \):

1. From \( 1 \):
- The coefficient of \( x^4 \) from \( (n+1)x^n \) is \( 5 \) (since \( n=4 \)).

2. From \( 2x \):
- The coefficient of \( x^3 \) from \( (n+1)x^n \) is \( 4 \) (since \( n=3 \)).
- Contribution: \( 2 \cdot 4 = 8 \).

3. From \( x^2 \):
- The coefficient of \( x^2 \) from \( (n+1)x^n \) is \( 3 \) (since \( n=2 \)).
- Contribution: \( 1 \cdot 3 = 3 \).

### Step 6: Add the contributions
Now, we sum all the contributions:
\[ 5 + 8 + 3 = 16 \]

Thus, the coefficient of \( x^4 \) in the expansion of \( \left( \frac{1+x}{1-x} \right)^2 \) is \( \boxed{16} \).

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