Divergence and convergence question

[Teachme]

Is the sum of 2 divergent series Ʃ(an±bn) divergent? From what I have learned is that it is not always divergent. Is this true? I believe that is what the picture i included is saying, but i maybe miss interpreting it. Also, is the product of 2 divergent series divergent or convergent?

 

[mathman]

The simplest example for the sum is an = -bn.

For the product case let an = bn = 1/n. ∑1/n diverges, ∑1/n² converges.

 

[Teachme]

Sorry I don't quite understand the sum example. Is the sum of 2 divergent series always divergent? Yes or no? Thanks.

 

[mathman]

The sum of two divergent series may be convergent or divergent. My example is a case of convergent. It is easy to construct other cases (an=bn) where it is divergent.

 

[Teachme]

Oh i understand the sum part, but I think the product of two divergent series is also divergent.
Because when you say that Ʃ1/n*Ʃ1/n is convergent you are missing an important rule Ʃan*Ʃbn≠ Ʃ(an*bn) so you can't apply your second example to the product of two divergent series. For multiplication of series is are like multiplying infinite quantities
such as (a_1+a_2+a_3)*(b_1+b_2+b_3) you have to distribute each term to each term and simplify. Thus Ʃan*Ʃbn≠ Ʃ(an*bn), which you made the assumption in your example.


I could be wrong, but this is what I have read.

 

[mathman]

If you are taking a product of two divergent series, it will always be divergent. However I thought you meant term by term product, not simply taking two numbers (the sums) and multiplying together.