En=1 (-1)^(n² + 1) 4 n

2. Prove the limit ratio test (Corollary 3.8 in the notes): Suppose {an} is a sequence of nonzero real numbers and r(n) = |an+1|. Suppose further that lim,-0 r (n) = L. Show that (i) If L 1, the series an diverges. Hint. This is a Corollary to the ratio test (Theorem 3.6 in the notes), so you should be using the ratio test in your proof.

IF in a normed space X, the absolute convergence of every series always implies convergence of the series show that X is a Banach space.

4. Prove the following corollary to the root test: If p(n) = |anl/" converges to a limit L, then: • If L 1 then the series diverges.