What condition on \(a_n\) makes \(\sum a_n\) a positive series in this topic?

If \(a_n\ge0\), what inequality relates \(S_{N+1}\) and \(S_N\)?

State whether \(\sum_{n=1}^{\infty}1/n^2\) has non-negative terms.

Explain why the partial sums of \(\sum_{n=1}^{\infty} e^{-n}\) are increasing.

Find \(S_3\) for the positive series \(\sum_{n=1}^{\infty}1/n^2\).

If \(0\le S_N\le 7\) for every \(N\) and \(S_N\) is increasing, what can you conclude?

Explain why \(\sum_{n=1}^{\infty}n^2\) diverges as a positive series.

For \(a_n=1/(n^2+1)\), show the partial sums are increasing.

Use comparison to show \(\sum_{n=1}^{\infty}1/2^n\) is bounded above.

Show that \(\sum_{n=1}^{\infty}1\) diverges using positive-series partial sums.

Explain why increasing partial sums do not automatically mean divergence.

Why does proving \(S_N\le M\) for all \(N\) matter for a positive series?

Show that if \(0\le a_n\le 1/2^n\), then \(\sum a_n\) converges.

Suppose \(a_n\ge 1/n\) for all \(n\). What can you conclude about \(\sum a_n\) if all terms are non-negative?

For \(a_n=p^n\) with \(p\ge0\), determine for which \(p\) the positive series \(\sum_{n=1}^{\infty}a_n\) converges.

If \(a_n\ge0\) and the partial sums satisfy \(S_{2N}\le4\) for every \(N\), prove the whole series converges.

A detector adds positive noise energies \(E_n\ge0\). If \(\sum_{n=1}^{N}E_n\le0.05\) for every \(N\), what does this imply?

Diagnose the error: positive partial sums increase, so every positive series diverges.

Prove that a positive series cannot have decreasing partial sums unless all new terms are zero.

A positive series has partial sums \(S_N=2-1/N\). Explain why this is consistent with convergence but cannot describe partial sums starting at \(N=1\) for strictly positive terms.