Let {ak) and [bk) be real sequences. Decide which of the following statements are true and which are false. Prove the true ones and give counterexamples to the false ones.
a) If ak ↓ 0, as k → ∞, and ∑∞k=1 bk converges conditionally, then ∑∞k=1 akbk converges.
b) If ak → 0, as k → ∞, then ∑∞k=1 (- 1)k ak converges.
c) If ak → 0, as k → ∞, and ak > 0 for all k ∈ N, then ∑∞k=1 (-1)k ak converges.
d) If ak → 0, as k → ∞, and ∑∞k=1 (- 1)k ak converges, then ak ↓ 0 as k → ∞.