If I have two series:
(1) 1/x₁ +1/x₂ +...
(2) 1/y₁ +1/y₂ +...

They both converge.

Does the series:
(3) 1/(y₁ x₁ )+1/(y₂ x₂ )+...
converge?

If I know that (1) converges to L and that (2) Converges two P, To what (3) converges (if it does) and is it expressable by P and L?

Thanks,
Yatir

This depends on whether you restrict x_n, y_n to being positive. If you do make this restriction then since

å

1/xi

converges then 1/x_i® 0 as i®¥. In particular, there exists n such that for all i > n we have 1/x_i < 1. This means that for all i > n we have 1/(y_i x_i) < 1/y_i and so 1/(y_n+1 x_n+1)+1/ (y_n+2 x_n+2)+¼ converges so

å

1/(yi xi)

converges.

If you don't restrict x_i, y_i to being positive and only restrict them to being non-zero, then the result is false.

For example set x_i=y_i=(-1)ⁱÖi then

å

1/xi

and

å

1/yi

converge (by the alternating series test) yet

å

1/(xi yi)

diverges.

Finally, even if

å

1/(xi yi)

converges, you cannot express this in terms of P and L. For example take x_i=y_i=1/2ⁱ then P=L=1 and the sum in (3) is 1/3. However if you take x_i=y_i=3/(4ⁱ) then P=L=1 still yet the sum in (3) is 51/17 or something, which is not 1/3. In other words even if the values of P and L are kept the same, it is possible for the sum in (3) to change.

Thanks Michael

Yatir