https://file.aiquickdraw.com/mathbot/file/1727854452187v7b3gbcx.jpg

The given image shows the infinite series:

\[
\sum_{n=1}^{\infty} \frac{\sin^2(2^n)}{n^4}
\]

This series involves a combination of trigonometric functions and powers of \(n\). To analyze this, let's break it down:

### 1. Behavior of the Numerator:
The term \(\sin^2(2^n)\) is bounded between 0 and 1 because for any real number \(x\), \(\sin^2(x)\) takes values in the range \([0, 1]\). Hence, the numerator does not grow significantly as \(n\) increases.

### 2. Denominator Growth:
The term \(n^4\) grows very rapidly as \(n\) increases. The factor \(n^4\) in the denominator suggests that the terms of the series decay quickly, meaning the series has a good chance of converging.

### 3. Comparison Test for Convergence:
We can use a comparison test with a simpler series. Consider the series:

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

This is a p-series with \(p = 4\), and it is known to converge since \(p > 1\). Because \(\sin^2(2^n)\) is bounded between 0 and 1, we have:

\[
0 \leq \frac{\sin^2(2^n)}{n^4} \leq \frac{1}{n^4}
\]

Thus, by the **comparison test**, since \(\sum_{n=1}^{\infty} \frac{1}{n^4}\) converges, the original series also converges.

### Conclusion:
The series \(\sum_{n=1}^{\infty} \frac{\sin^2(2^n)}{n^4}\) **converges**.

Would you like more details on how to compute the exact sum, or any specific aspects of this problem? 

Here are some questions you might explore further:
1. How does the rapid growth of \(n^4\) ensure convergence in series?
2. Can we approximate the sum of the series numerically?
3. What if the power in the denominator was \(n^2\) instead of \(n^4\)?
4. How do different types of tests help in determining convergence of series?
5. What are other examples of convergent series involving trigonometric functions?

**Tip:** When testing for convergence, comparison tests are very useful when you can bound the terms of your series with a simpler, known series.

Evaluate the convergence of the infinite series involving trigonometric and polynomial terms: ∑ (sin²(2ⁿ) / n⁴) from n=1 to ∞.

Analyze the convergence of the infinite series ∑ (sin²(2ⁿ) / n⁴) from n=1 to ∞ using the Comparison Test and p-series theorem. This problem illustrates key trigonometric and series concepts suitable for undergraduate students.

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