We are given the sequence $a_n = \sqrt[n]{n}$, and we are tasked with computing the first six terms, rounding the results to four decimal places.

The first six terms:

1. $a_1 = \sqrt[1]{1} = 1.0000$
2. $a_2 = \sqrt[2]{2} = \sqrt{2} \approx 1.4142$
3. $a_3 = \sqrt[3]{3} \approx 1.4422$
4. $a_4 = \sqrt[4]{4} = \sqrt[2]{2} \approx 1.1892$
5. $a_5 = \sqrt[5]{5} \approx 1.1487$
6. $a_6 = \sqrt[6]{6} \approx 1.1225$

Convergence of the Sequence:

We are asked whether the sequence converges or diverges. As $n \to \infty$, $\sqrt[n]{n}$ approaches 1 because the root grows faster than the value of $n$, which causes the expression to get closer to 1. Hence, the sequence converges to 1.

So, the limit is $\boxed{1}$.

Would you like more details or have any questions?

Relative Questions:

1. What is the general behavior of $\sqrt[n]{n}$ as $n$ increases?
2. How can we prove that $\lim_{n \to \infty} \sqrt[n]{n} = 1$ rigorously?
3. How does the sequence $\sqrt[n]{n}$ compare to other common sequences, like $\frac{1}{n}$?
4. Can you find another sequence where the root increases faster than the base, leading to divergence?
5. What happens to the sequence if $n$ grows extremely large, say in the thousands?

Tip:

Sequences involving roots tend to converge slowly. Understanding how different parts of the expression behave (like the base and the root) helps in analyzing limits more intuitively.