State the \(p\)-series convergence rule for \(\sum_{n=1}^{\infty}1/n^q\).

What does \(\rho=1\) mean in the ratio test?

Use the \(p\)-series rule to classify \(\sum_{n=1}^{\infty}1/n^3\).

Use the term test on \(\sum_{n=1}^{\infty}\frac{2n+1}{n}\).

Apply the ratio test to \(\sum_{n=1}^{\infty}1/n!\).

Apply the root test to \(\sum_{n=1}^{\infty}(1/5)^n\).

Use comparison to show \(\sum_{n=1}^{\infty}1/(n^2+4)\) converges.

Use comparison to show \(\sum_{n=1}^{\infty}\frac{n+1}{n}\) diverges.

Apply the ratio test to \(\sum_{n=1}^{\infty} n^2/2^n\).

Use limit comparison with \(1/n^2\) for \(\sum_{n=1}^{\infty}\frac{3n+1}{n^3+2}\).

Explain why the integral test requires positivity and decreasing behavior on the tail.

A ratio test gives \(\rho=3/2\). What conclusion follows, and why?

Use the root test on \(\sum_{n=1}^{\infty}\left(\frac{n}{2n+1}\right)^n\).

Use the integral test to classify \(\sum_{n=2}^{\infty}1/(n\ln n)\).

For which real \(p\) does \(\sum_{n=1}^{\infty}1/n^{2p}\) converge?

Classify \(\sum_{n=1}^{\infty}\frac{5^n}{n!}\) using the ratio test.

Classify \(\sum_{n=1}^{\infty}\frac{n!}{3^n}\) using the ratio test.

Diagnose the error: the ratio test gives \(\rho=1\), so the series converges.

Choose and justify a test for \(\sum_{n=1}^{\infty}\frac{2n^2+1}{n^4+n}\).

A learner tries the ratio test on \(\sum1/n^2\) and gets \(\rho=1\). Give a better conclusion and method.