Coffiecient of `x^3` in the expansion of `e^(-bx)`

Options

A. `(-b^3)/6`
B. `(b^3)/6`
C. `(b^3)/3`
D. `(-b^3)/3`

Answer

To find the coefficient of \( x^3 \) in the expansion of \( e^{-bx} \), we can use the Taylor series expansion of \( e^x \).

### Step-by-step Solution:

1. **Recall the Taylor Series Expansion**:
The Taylor series expansion of \( e^x \) is given by:
\[ e^x = 1 + \frac{x}{1!} + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots \]
For \( e^{-bx} \), we replace \( x \) with \( -bx \):
\[ e^{-bx} = 1 + \frac{-bx}{1!} + \frac{(-bx)^2}{2!} + \frac{(-bx)^3}{3!} + \cdots \]

2. **Expand the Series**:
Now, we can expand the series:
\[ e^{-bx} = 1 - \frac{bx}{1} + \frac{b^2x^2}{2} - \frac{b^3x^3}{6} + \cdots \]

3. **Identify the Coefficient of \( x^3 \)**:
From the expansion, we can see that the term involving \( x^3 \) is:
\[ -\frac{b^3x^3}{6} \]
Therefore, the coefficient of \( x^3 \) is:
\[ -\frac{b^3}{6} \]

4. **Conclusion**:
Thus, the coefficient of \( x^3 \) in the expansion of \( e^{-bx} \) is:
\[ -\frac{b^3}{6} \]

### Final Answer:
The coefficient of \( x^3 \) in the expansion of \( e^{-bx} \) is \( -\frac{b^3}{6} \).

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