The given summation involves a series of terms in the form: 

\[
\sum_{n=1}^{\infty} \frac{1}{n^4}.
\]

This is a **special mathematical series** known as the **Riemann zeta function**, evaluated at \( s = 4 \). The Riemann zeta function is defined as:

\[
\zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s}.
\]

For \( s = 4 \):

\[
\zeta(4) = \sum_{n=1}^{\infty} \frac{1}{n^4}.
\]

### Value of \( \zeta(4) \):
Using advanced mathematical results, \( \zeta(4) \) can be expressed in terms of \(\pi\):

\[
\zeta(4) = \frac{\pi^4}{90}.
\]

### Approximation:
Substituting \( \pi \approx 3.14159 \), we get:

\[
\zeta(4) = \frac{(3.14159)^4}{90} \approx \frac{97.409}{90} \approx 1.08232.
\]

Thus:

\[
\sum_{n=1}^{\infty} \frac{1}{n^4} \approx 1.08232.
\]

### Explanation:
This result converges quickly because \( \frac{1}{n^4} \) decreases rapidly as \( n \) grows.

---

Would you like a detailed derivation or more explanation on the Riemann zeta function? 

Here are 5 related questions for further exploration:
1. What is the Riemann zeta function and its importance in mathematics?
2. How is the formula for \( \zeta(4) \) derived using Fourier series or integrals?
3. What are the values of \( \zeta(s) \) for other even integers, like \( \zeta(2) \)?
4. What is the relationship between the Riemann zeta function and number theory?
5. How does the convergence of the series \( \sum_{n=1}^\infty \frac{1}{n^s} \) depend on \( s \)?

### Tip:
For any positive integer \( s = 2k \) (even), \( \zeta(s) \) can be calculated exactly using a formula involving Bernoulli numbers and \(\pi\).

Explore how to evaluate the infinite series ∑ 1/n^4, a special case of the Riemann zeta function at s=4. Learn about the formula ζ(4) = π^4 / 90 and its approximate value 1.08232. Ideal for advanced mathematics learners in Grades 11-12 and beyond.

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