# Math Help - How do I show the series of an*bn converges?

1. ## How do I show the series of an*bn converges?

How do I show the series of an*bn converges?

2. $|ab|\le a^2+b^2$

0<=(a-b)^2
2ab<=a^2+b^2
2ab<=a^2+b^2
therefore

2*sum(ab)<=sum(a^2)+sum(b^2)

Where the RHS converges

As the LHS converges absolutely sum(ab) converges?

this is from a past paper by the way, I can link you if you like, it's not a take home test

4. hint:

Consider the series

$a_n=\left( \frac{i}{\sqrt{n}} \right)^n$

Notice that

$(a_n)^2 = \frac{(-1)^n}{n}$

$b_n=\left( \frac{-i}{\sqrt{n}} \right)^n$

$(b_n)^2=\frac{(-1)^n}{n}$

but ...

5. Ah, but an*bn=1/n.. and taking the the sum of 1/n we find by the integral test that is diverges! And by the Alternating series test (-1)^n/n converges
Thanks!