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

What does absolute convergence imply about ordinary convergence?

Write the absolute-value series for \(\sum_{n=1}^{\infty}(-1)^n/n^2\).

Write \(|(-3)^n/5^n|\) in simpler form.

Determine whether \(\sum_{n=1}^{\infty}(-1)^n/n^2\) converges absolutely.

Determine whether \(\sum_{n=1}^{\infty}(-1)^n/2^n\) converges absolutely.

Use comparison to test absolute convergence of \(\sum (-1)^n/(n^2+1)\).

Use the ratio test on the absolute-value series for \(\sum (-1)^n n!/4^n\).

Test absolute convergence of \(\sum_{n=1}^{\infty}(-1)^n\frac{n}{3^n}\).

Test absolute convergence of \(\sum_{n=1}^{\infty}\sin(n)/n^2\).

Explain why divergence of \(\sum |a_n|\) does not by itself prove divergence of \(\sum a_n\).

Why are rearrangements safe for absolutely convergent series?

For which real \(p\) does \(\sum (-1)^n/n^p\) converge absolutely?

Test absolute convergence of \(\sum_{n=1}^{\infty}(-1)^n\frac{2^n}{n^3}\).

For which real \(r\) does \(\sum_{n=0}^{\infty}(-1)^n r^n\) converge absolutely?

Show absolute convergence of \(\sum_{n=1}^{\infty}(-1)^n/(n^2+n)\) by comparison.

A correction series has signed terms \(a_n=(-1)^n0.2^n/n\). Prove the total magnitude is finite.

Diagnose the error: \(\sum a_n\) converges, so \(\sum |a_n|\) must converge.

Prove that \(\sum |a_n|\) convergent implies \(\sum a_n\) convergent using positive and negative parts conceptually.

A learner tests \(\sum (-1)^n n/2^n\) by saying the signs alternate, so it converges. Give a stronger correct conclusion.