1. Prove the ratio comparison test: If a n > 0 and b n > 0 for all n, if Σa n converges, and if b n+1 /b n ≤ a n + 1/a n for all n, then Σb n converges 2. (a) Let Σa n and Σb n be two series of...

1. Prove the ratio comparison test: If a_n
> 0 and b_n
> 0 for all n, if Σa_n
converges, and if b_n+1/b_n
≤ a_n
+ 1/a_n
for all n, then Σb_n
converges

2. (a) Let Σa_n
and Σb_n
be two series of positive terms and suppose that the sequence (a_n/b_n) converges to a nonzero number. Prove that Σa_n
converges iff Σb_n
converges. (This is sometimes called the limit comparison test.)

(b) Suppose a_n
≥ 0 and b_n
> 0 for all n. If lim sup (a_n/b_n) is finite and Σb_n
converges, then Σa_n
converges