Are the series absolutely, conditionally convergent, or divergent?

Step-by-step explanation:

∑ sin(nπ/4) / (n³ + 3n)

This is less than 1 / n⁴ for all n > 1.  1/n⁴ is a convergent p-series, so the lesser series also converges.  │aₙ│converges for the same reason, so this is absolutely convergent.

∑ (-1)ⁿ⁺¹ ln(n + 1) / (n + 1)

This is an alternating series.  bₙ is positive and decreasing, and lim(n→∞) bₙ = 0, so the series converges.  Now we need to check if│aₙ│converges.  Using comparison test, │aₙ│is greater than 1/n for all n ≥ 6.  1/n is a divergent p-series, so the greater series│aₙ│also diverges.  So this is conditionally convergent.

hm let me think hm think think hm think yes