We are tasked with finding the limit of the sequence $\{ \tan^{-1} \left( \frac{10n}{10 + 4n} \right) \}$ as $n \to \infty$, or showing that the sequence diverges.

Step 1: Analyze the expression inside the arctangent

The sequence can be rewritten as:

$a_n = \tan^{-1} \left( \frac{10n}{10 + 4n} \right)$

Step 2: Simplify the fraction

We can simplify the fraction inside the arctangent:

$\frac{10n}{10 + 4n}$

Dividing the numerator and the denominator by $n$, we get:

$\frac{10n}{10 + 4n} = \frac{10}{\frac{10}{n} + 4}$

As $n \to \infty$, $\frac{10}{n} \to 0$, so the expression simplifies to:

$\frac{10}{4} = 2.5$

Step 3: Consider the behavior of the sequence as $n \to \infty$

Thus, as $n \to \infty$, the argument of $\tan^{-1}$ approaches 2.5. Since $\tan^{-1}(x)$ is a continuous function, we can pass the limit inside the arctangent:

$\lim_{n \to \infty} a_n = \tan^{-1}(2.5)$

Step 4: Conclusion

Therefore, the sequence converges to $\tan^{-1}(2.5)$, which is a finite value.

Hence, the limit of the sequence is:

$\lim_{n \to \infty} a_n = \tan^{-1}(2.5)$

This shows that the sequence converges.

Would you like more details on arctangent limits, or any further explanation?

---

Here are 5 related questions:

1. How do you compute $\tan^{-1}(x)$ for a given $x$?
2. What is the behavior of the function $\tan^{-1}(x)$ as $x \to \infty$?
3. Can sequences involving trigonometric functions diverge? Under what conditions?
4. What are the properties of continuous functions when applied to limits of sequences?
5. How would the limit change if the sequence were $\tan^{-1} \left( \frac{10n}{10 + n} \right)$?

Tip: When simplifying fractions in limits, always check if dividing both numerator and denominator by the highest power of $n$ helps to clarify the limit behavior.