What feature distinguishes a mixed-sign series from a positive series?

What is \(|a_n|\) used for when studying signed series?

Write the first four terms of \(\sum_{n=1}^{\infty}(-1)^{n+1}/n\).

State whether \(\sum_{n=1}^{\infty}(-1)^n/n^2\) is a positive series.

Find the magnitudes of the first four terms of \(-1+1/2-1/3+1/4-\cdots\).

If only \(a_1\) and \(a_2\) are negative and all later terms are non-negative, what part matters for convergence?

Explain why \(\sum(-1)^n\) diverges using partial sums.

For \(a_n=(-1)^n/(n+1)\), write \(|a_n|\).

Why can cancellation make a signed series converge when the corresponding positive intuition is not enough?

Show why positive-series comparison cannot be applied directly to \(\sum (-1)^{n+1}/n\).

Explain why finitely many negative terms do not affect convergence of an otherwise non-negative series.

A charge model sums \(+1,-1/2,+1/3,-1/4,\ldots\). What sign feature must be analyzed before using positive-series tests?

Classify the sign pattern of \(a_n=(-1)^{n+1}/(n^2+1)\) and give the magnitude sequence.

If \(a_n=(-1)^n n/(n+1)\), use the term condition to classify \(\sum a_n\).

A signed series has negative terms only when \(n=1,2,3\), and for \(n\ge4\), \(a_n=1/n^2\). Does it converge?

Give an example of a mixed-sign series that diverges because its terms do not tend to zero, and justify it.

A signed displacement correction has terms \((-1)^{n+1}0.1^n\). Why is it reasonable to test magnitudes first?

Diagnose the error: \(\sum (-1)^{n+1}/n\) diverges because \(\sum1/n\) diverges.

Explain why rearranging a conditionally convergent mixed-sign series is risky.

Prove that if \(\sum |a_n|\) converges for a signed series, then \(a_n\to0\).