Answer:
Explanation:
The root test is a convergence test used to determine whether an infinite series converges or diverges by considering the limit of the nth root of the absolute value of the terms in the series. If the limit is less than 1, the series converges; if it's greater than 1, the series diverges; if it's equal to 1, the test is inconclusive.
Let's apply the root test to each of the given series:
1. \(\sum_{n=1}^\infty \left(\frac{\pi}{2\arctan n}\right)^n\)
- Apply the root test:
\[\lim_{n\to\infty}\sqrt[n]{\left|\frac{\pi}{2\arctan n}\right|^n} = \frac{\pi}{2}\lim_{n\to\infty}\frac{1}{\arctan n}\]
The limit here is greater than 1, so the root test indicates divergence.
2. \(\sum_{n=1}^\infty \left(\frac{n}{\ln n}\right)^{\frac{n}{2}}\)
- Apply the root test:
\[\lim_{n\to\infty}\sqrt[n]{\left|\left(\frac{n}{\ln n}\right)^{\frac{n}{2}}\right|} = \lim_{n\to\infty}\frac{n}{\ln n}\]
The limit here is greater than 1, so the root test indicates divergence.
3. \(\sum_{n=1}^\infty \left(\frac{3n-1}{3n}\right)^{n^2}\)
- Apply the root test:
\[\lim_{n\to\infty}\sqrt[n]{\left|\left(\frac{3n-1}{3n}\right)^{n^2}\right|} = \lim_{n\to\infty}\frac{3n-1}{3n}\]
The limit here is less than 1, so the root test indicates convergence.
4. \(\sum_{n=1}^\infty \left(\frac{2\arctan n}{\pi}\right)^n\)
- Apply the root test:
\[\lim_{n\to\infty}\sqrt[n]{\left|\left(\frac{2\arctan n}{\pi}\right)^n\right|} = \frac{2}{\pi}\lim_{n\to\infty}\arctan n\]
The limit here is greater than 1, so the root test indicates divergence.
5. \(\sum_{n=1}^\infty \left(\frac{2n^2+4n+1}{3n^2+1}\right)^n\)
- Apply the root test:
\[\lim_{n\to\infty}\sqrt[n]{\left|\left(\frac{2n^2+4n+1}{3n^2+1}\right)^n\right|} = \lim_{n\to\infty}\frac{2n^2+4n+1}{3n^2+1}\]
The limit here is less than 1, so the root test indicates convergence.
6. \(\sum_{n=1}^\infty \left(\frac{2n}{2n+1}\right)^{2n}\)
- Apply the root test:
\[\lim_{n\to\infty}\sqrt[n]{\left|\left(\frac{2n}{2n+1}\right)^{2n}\right|} = \lim_{n\to\infty}\frac{2n}{2n+1}\]
The limit here is greater than 1, so the root test indicates divergence.
7. \(\sum_{n=1}^\infty \left(\frac{\ln(3n+1)}{n}\right)^{2n}\)
- Apply the root test:
\[\lim_{n\to\infty}\sqrt[n]{\left|\left(\frac{\ln(3n+1)}{n}\right)^{2n}\right|} = \lim_{n\to\infty}\frac{\ln(3n+1)}{n}\]
The limit here is less than 1, so the root test indicates convergence.
8. \(\sum_{n=1}^\infty \left(\frac{3n}{3n-1}\right)^{n^2}\)
- Apply the root test:
\[\lim_{n\to\infty}\sqrt[n]{\left|\left(\frac{3n}{3n-1}\right)^{n^2}\right|} = \lim_{n\to\infty}\frac{3n}{3n-1}\]
The limit here is greater than 1, so the root test indicates divergence.
9. \(\sum_{n=1}^\infty \left(\frac{n^4\