give an example of two divergent sequences X and Y such that
a. X + Y converges
b. XY converges

Question

give an example of two divergent sequences X and Y such that 
a. X + Y converges 
b. XY converges

jamesj · Answer

Let X = (x_n), x_n = n and Y, y_n = -n.  Then both X and Y diverge but X + Y converges because x_n + y_n = 0 for all n.

jamesj · Answer

that's one of them.  Now I'll let you think about the second one a bit.

anonymous · Answer

Please clarify the notation

jamesj · Answer

X is the sequence 1, 2, 3, 4, 5 ....
Y is the sequence -1, -2, -3, -4, -5, ...

anonymous · Answer

thnx

jamesj · Answer

$$X = (x_n) \ \text{ where } \ x_n = n$$
Hence
$$x_1 = 1, \ \ x_2 = 2, \ \  x_3 = 3, \ ...$$

anonymous · Answer

-n is the subscript got it

jamesj · Answer

And remember, a sequence is divergent just means it is not convergent.  It's not that the sequence has to go off to infinity like the examples I just gave.  It could just alternate up and down for instance.

jamesj · Answer

e.g., x_n = (-1)^n is a divergent sequence.

jamesj · Answer

And that's a big hint for part b.

anonymous · Answer

because (-1)^n and the product of its reciprocal is equal to 1. hence (-1)^n and 1/((-1)^n) is 1

jamesj · Answer

I'm not quite following you; are you missing a post where you defined your X and Y?