The problem you presented involves a series of expressions, including a logarithmic integral, a trigonometric sum, and a famous series related to the Riemann zeta function. Let's break it down step by step:

Step 1: Clarification of Expressions

1. $\ln(0, 2\pi)$: This part isn't clear, as the logarithmic function is undefined at 0. It seems like you're referring to a domain or an integral involving logarithmic functions. However, without further context, I can't interpret this directly. Could you clarify this part?

2. $\left( \frac{\pi - x}{2} \right)^2 = \frac{\pi^2}{12} + \sum_{n=1}^{\infty} \frac{\cos(nx)}{n^2}$:

   • This appears to be a Fourier series expansion, particularly related to a known trigonometric series.
   • This form is related to the Fourier expansion of a function, and it's likely connected to the series expansion of the square of a sine function.

Step 2: Evaluation of the Series

Next, let's focus on the infinite sum you mentioned at the end: $\frac{1}{1^4} + \frac{1}{2^4} + \frac{1}{3^4} + \frac{1}{4^4} + \cdots$

This is the Riemann zeta function $\zeta(s)$ evaluated at $s = 4$: $\zeta(4) = \sum_{n=1}^{\infty} \frac{1}{n^4}$ The value of $\zeta(4)$ is a well-known mathematical constant: $\zeta(4) = \frac{\pi^4}{90}$ Hence, $\sum_{n=1}^{\infty} \frac{1}{n^4} = \frac{\pi^4}{90} \approx 1.08232$

Summary:

• The infinite sum $\frac{1}{1^4} + \frac{1}{2^4} + \frac{1}{3^4} + \cdots$ evaluates to $\zeta(4) = \frac{\pi^4}{90}$.
• For the other parts of the question, further clarification (especially about the logarithmic term) might be needed to provide a complete solution.

Would you like more details on the Fourier series or any other part of this?

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Here are 5 related questions to consider:

1. What is the significance of the zeta function for even integers?
2. Can you prove the value of $\zeta(4) = \frac{\pi^4}{90}$?
3. How does Fourier series expansion work in general?
4. What is the relationship between trigonometric sums and zeta functions?
5. Can the logarithmic integral be applied to physical systems, and how?

Tip: The Riemann zeta function at even integers yields results involving powers of $\pi$, and it connects to many areas of number theory and analysis.