Okay so, the limit as n goes to infinity of a_n/b_n = L >0; If sum of b_n converges then the sum of a_n also converges and if sum of b_n diverges then sum of a_n diverges.

Zero-Case: If the limit as n goes to infinity of a_n/b_n = 0; if sum of b_n converges then the sum of a_n also converges. Is it true that: the limit as n goes to infinity of a_n/b_n = 0; if the sum of b_n diverges then the sum of a_n also diverges? Is it also true that: the limit as n goes to infinity of b_n/a_n = 0; if the sum of b_n converges (or diverges) then the sum of a_n converges (or diverges)?

Infinity-Case: if the limit as n goes to infinity of a_n/b_n = infinity; if the sum of b_n converges then the sum of a_n converges. Is this true: if the sum of b_n diverges then the sum of a_n also diverges? Is it also true that: the limit as n goes to infinity of b_n/a_n = infinity; if the sum of b_n converges (or diverges) then the sum of a_n converges (or diverges)?

Can someone confirm or correct me on these statements, please?