The image presents a mathematical problem involving a series and an equation, which appears to be related to Fourier series or trigonometric sums.

The given expression is: $\ln(0, 2\pi)\left(\frac{\pi - x}{2}\right)^2 = \frac{\pi^2}{12} + \sum_{n=1}^{\infty} \frac{\cos(nx)}{n^2}$ It asks to evaluate the following sum: $\sum_{n=1}^{\infty} \frac{1}{n^4} = \frac{1}{1^4} + \frac{1}{2^4} + \frac{1}{3^4} + \frac{1}{4^4} + \cdots$

The given options are:

• (a) $\frac{\pi^4}{90}$
• (b) $\frac{\pi^4}{96}$
• (c) $\frac{\pi^4}{92}$
• (d) $\frac{\pi^4}{94}$

Solution:

The series in the problem is the Riemann zeta function evaluated at $s = 4$: $\zeta(4) = \sum_{n=1}^{\infty} \frac{1}{n^4}$ It is known that: $\zeta(4) = \frac{\pi^4}{90}$

Thus, the value of the sum is $\frac{\pi^4}{90}$, which matches option (a).

Final Answer:

The correct answer is (a) $\frac{\pi^4}{90}$.

Would you like further details or clarification on any part of the solution?

Here are 5 relative questions that expand the previous explanation:

1. What is the Riemann zeta function, and how is it generally defined?
2. Can you explain how the value of $\zeta(4)$ is derived?
3. What other important values of the Riemann zeta function are commonly used in mathematical problems?
4. How are Fourier series related to trigonometric sums like the one shown in the equation?
5. Can this approach be extended to other sums involving higher powers of $n$?

Tip: The Riemann zeta function has many important applications in both pure and applied mathematics, particularly in number theory and physics.