It looks like the image contains part of a Fourier series problem or some similar trigonometric series expansion. The red text shows expressions involving sums, sine functions, and integrals. Here’s a breakdown of the visible parts:

1. Series Sum: $\sum n \cdot (A_n - B_n) \cdot \sin\left(\frac{n \pi x}{2}\right) = 15$ This is likely part of a Fourier series where $A_n$ and $B_n$ are Fourier coefficients, and $\sin\left(\frac{n \pi x}{2}\right)$ is a sine term. The equation is equal to 15, which might represent the value of the function being modeled by the Fourier series at a specific point.

2. Expression for $b_n$: $b_n = \frac{2}{L} \int_0^L 15 \cdot \sin\left(\frac{n \pi x}{2}\right) dx$ This looks like the formula for calculating the Fourier sine coefficients $b_n$, where $L = 2$ is the length of the interval. The integrand involves the product of the function value (here, 15) and a sine term. The bounds of the integral run from 0 to $L$.

3. Variables and Coefficients:

   • $L = 2$: The period or interval length for the Fourier series.
   • $A_n - B_n$: The difference between two Fourier coefficients (possibly cosine and sine coefficients).
   • $b_n$: Likely represents the Fourier sine coefficient.

Would you like me to solve the integral for $b_n$ or provide further clarification on any part of the problem?

Here are 5 related questions to further explore:

1. What are $A_n$ and $B_n$ in this context?
2. Are the sine terms part of a Fourier sine series or another expansion?
3. What is the significance of the value 15 in the equation?
4. Can we assume the function is periodic with period $L = 2$?
5. How are the coefficients $b_n$ used in reconstructing the original function?

Tip: Fourier coefficients $b_n$ are calculated by integrating over one period of the function, weighted by sine or cosine functions.