Question

The coefficient of \( x^4 \) in the power series expansion of \( e^{\sin x} \) about \( x = 0 \) is ............. (correct up to three decimal places).

Hint:

For series expansions, remember to substitute known series expansions and combine terms to find the desired coefficient.

Answer 1

Correct Answer: -0.13

Step 1: Series expansion of \( e^{\sin x} \).
The power series expansion of \( \sin x \) is: \[ \sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots \] Thus, the expansion of \( e^{\sin x} \) is: \[ e^{\sin x} = 1 + \sin x + \frac{\sin^2 x}{2!} + \frac{\sin^3 x}{3!} + \cdots \]

Step 2: Calculate the coefficient of \( x^4 \).
We only need terms up to \( x^4 \) in the expansion. By substituting the series of \( \sin x \) into the expansion of \( e^{\sin x} \) and collecting terms, we find that the coefficient of \( x^4 \) is approximately \( \boxed{-\frac{1}{6}} \).