Coffiecient of `x^n` in `e^(a +bx) ` is

Options

A. `(e^ab^n)/n`
B. `(b^n)/(n!)`
C. `(e^(b)a^n)/(n!) `
D. `(e^a b^n)/(n!)`

Answer

To find the coefficient of \( x^n \) in the expression \( e^{a + bx} \), we can follow these steps:

### Step 1: Rewrite the Expression
We start by rewriting the expression \( e^{a + bx} \) as:
\[ e^{a + bx} = e^a \cdot e^{bx} \]

### Step 2: Expand \( e^{bx} \)
Next, we use the Taylor series expansion for \( e^{bx} \):
\[ e^{bx} = \sum_{n=0}^{\infty} \frac{(bx)^n}{n!} = 1 + \frac{bx}{1!} + \frac{(bx)^2}{2!} + \frac{(bx)^3}{3!} + \ldots \]
This can be written as:
\[ e^{bx} = \sum_{n=0}^{\infty} \frac{b^n x^n}{n!} \]

### Step 3: Combine with \( e^a \)
Now, substituting this back into our expression, we have:
\[ e^{a + bx} = e^a \cdot \sum_{n=0}^{\infty} \frac{b^n x^n}{n!} \]

### Step 4: Identify the Coefficient of \( x^n \)
To find the coefficient of \( x^n \) in \( e^{a + bx} \), we look at the term in the series expansion:
\[ \text{Coefficient of } x^n = e^a \cdot \frac{b^n}{n!} \]

### Final Result
Thus, the coefficient of \( x^n \) in \( e^{a + bx} \) is:
\[ \boxed{e^a \cdot \frac{b^n}{n!}} \]

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