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 ∑∞k=1 ak converges and ak/bk → 0 as k → ∞, then ∑∞k=1 bk converges.
b) Suppose that 0 < a < l. If ak > 0 and k√ak for all k ∈ N, then ∑∞k=1 ak converges.
c) Suppose that ak → 0 as k ∞. If ak > 0 and √ak+1 < ak for all k ∈ N, then ∑∞k=1 ak converges.
d) Suppose that ak = f(k) for some continuous function f : [1, ∞) → [0, ∞) which satisfies f(x) → 0 as x → ∞. If ∑∞k=1 ak converges, then ∫∞1 f(x)dx converges.