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The image appears to contain a mathematical problem related to expanding a Fourier series. The text I can discern is:

**"Expand Fourier Series for \( f(x) = e^{x(1-x)} \)"**

To solve this, we need to express the function \( f(x) = e^{x(1-x)} \) as a Fourier series, which involves finding the coefficients of the series that represent the function in terms of sines and cosines.

### Step-by-Step Process for Expanding a Fourier Series:

1. **Identify the Period:**  
   The first step is to determine the period of the function. If not specified, we assume the function is defined over a typical interval, like \([0, 1]\) or \([- \pi, \pi]\). Since \( f(x) = e^{x(1-x)} \), we might need to consider its period depending on the given range.

2. **Determine the Fourier Coefficients:**
   The Fourier series of a function \(f(x)\) on the interval \([-L, L]\) is typically expressed as:
   \[
   f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty} \left( a_n \cos \left( \frac{n \pi x}{L} \right) + b_n \sin \left( \frac{n \pi x}{L} \right) \right),
   \]
   where:
   \[
   a_n = \frac{1}{L} \int_{-L}^{L} f(x) \cos \left( \frac{n \pi x}{L} \right) dx, \quad b_n = \frac{1}{L} \int_{-L}^{L} f(x) \sin \left( \frac{n \pi x}{L} \right) dx.
   \]

3. **Compute the Coefficients:**  
   Substitute the function \(f(x) = e^{x(1-x)}\) and compute the integrals for \(a_0\), \(a_n\), and \(b_n\).

Would you like me to go ahead with computing the Fourier coefficients for this function, or is there something specific you'd like to clarify? 

### Questions to Explore:
1. What is the period \(L\) for the given function \(f(x) = e^{x(1-x)}\)?
2. Should the Fourier series expansion be over a specific interval, like \([0, 1]\) or \([- \pi, \pi]\)?
3. Are there any conditions or constraints given for \(f(x)\) that I should consider?
4. Would you like a more detailed explanation of how to compute the Fourier coefficients?
5. Are you interested in understanding the convergence properties of the Fourier series?

### Tip:
When expanding functions into their Fourier series, always check if the function is even or odd; this can simplify the computation of coefficients.

Expand Fourier Series for f(x) = e^{x(1-x)}

Explore how to expand the Fourier series for the function f(x) = e^{x(1-x)}. This detailed guide covers finding Fourier coefficients and includes key steps for computing integrals over a given period. Suitable for students studying Fourier series in undergraduate math.

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