Define conditional convergence for \(\sum a_n\).

State the two magnitude conditions used in the alternating series test.

Write the absolute-value series for the alternating harmonic series \(\sum (-1)^{n+1}/n\).

Do the magnitudes \(1/n\) tend to zero and decrease?

Show that \(\sum_{n=1}^{\infty}(-1)^{n+1}/n\) satisfies the alternating series test.

Show that the alternating harmonic series is not absolutely convergent.

Determine whether \(\sum_{n=1}^{\infty}(-1)^n/n^2\) is conditionally convergent.

Determine whether \(\sum_{n=2}^{\infty}(-1)^n/\ln n\) passes the alternating series test.

Show that \(\sum_{n=1}^{\infty}(-1)^{n+1}/\sqrt n\) converges conditionally.

Explain why conditional convergence depends on cancellation.

Why does the alternating series test not prove absolute convergence?

A series converges by the alternating series test, and its absolute-value series also converges. Is it conditional?

For which real \(p\) is \(\sum_{n=1}^{\infty}(-1)^{n+1}/n^p\) conditionally convergent?

Test \(\sum_{n=1}^{\infty}(-1)^n n/(n+1)\) for conditional convergence.

For which real \(p\) is \(\sum_{n=2}^{\infty}(-1)^n/(n(\ln n)^p)\) conditionally convergent, assuming the alternating test applies?

A signed approximation has terms \((-1)^{n+1}/n\). What warning does conditional convergence give about rearranging corrections?

Show that \(\sum_{n=1}^{\infty}(-1)^n\frac{\ln n}{n}\) is not absolutely convergent, and state the ordinary convergence idea.

Diagnose the error: the alternating harmonic series converges, so its absolute-value series must converge too.

Prove that conditional convergence requires two separate tests, not one.

A learner says every alternating series with terms tending to zero is conditionally convergent. Correct the statement with examples.