(a) Suppose that the sequence (cn) converges to zero. Show that ∑n≥0cn converges if and only if ∑k≥0(c2k+c2k+1) converges. Moreover, if the two series converge then they have the same limit.
(b) Give an example where (cn) does not converge to 0 and one of the series in (a) diverges while the other converges.