∑an和∑bn都发散,那么∑an*bn和∑an/bn都发散吗?怎么证明?

Question

∑an和∑bn都发散,那么∑an*bn和∑an/bn都发散吗?怎么证明?

Answer 1

两个答案都是不一定.

比如：
an={0,1,0,1,0,1,...},
bn={1,0,1,0,1,0,...},所以∑an和∑bn都发散,但是an*bn恒等于0,即∑an*bn不发散.

又比如,an=1,bn=2ⁿ,那么∑an和∑bn都发散,但是∑an/bn是等比数列,极限是1,不发散.