Coefficient of `x^(10)` in `e^(3x)` is

Options

A. `3^(10)`
B. `3^(10)/(10!)`
C. `3^(10)/(10)`
D. `3^(10)/(10!)`

Answer

To find the coefficient of \( x^{10} \) in the expansion of \( e^{3x} \), we can use the Taylor series expansion of the exponential function. The series expansion of \( e^x \) is given by:

\[ e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!} \]

By substituting \( 3x \) for \( x \), we have:

\[ e^{3x} = \sum_{n=0}^{\infty} \frac{(3x)^n}{n!} = \sum_{n=0}^{\infty} \frac{3^n x^n}{n!} \]

To find the coefficient of \( x^{10} \), we need to identify the term in the series where \( n = 10 \):

\[ \text{Term for } n = 10: \frac{3^{10} x^{10}}{10!} \]

Thus, the coefficient of \( x^{10} \) in \( e^{3x} \) is:

\[ \frac{3^{10}}{10!} \]

### Final Answer:
The coefficient of \( x^{10} \) in \( e^{3x} \) is \( \frac{3^{10}}{10!} \).

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