Show that a series converges absolutely given convergence of two related series

Show that if and are convergent then is absolutely convergent.

Hint: Show |xy| <= 1/2(|x|^2 + |y|^2) for any x,y ε R

I can prove the Hint -

And this is true for all x, y as the square of any real number will be greater than or equal to 0.

But I cant see how to apply this to the original question?

Re: Show that a series converges absolutely given convergence of two related series

Quote:

Originally Posted by

**tomcruisex** Show that if

and

are convergent then

is absolutely convergent.

Hint: Show |xy| <= 1/2(|x|^2 + |y|^2) for any x,y ε R

I can prove the Hint -

And this is true for all x, y as the square of any real number will be greater than or equal to 0.

But I cant see how to apply this to the original question?

First notice that and .

Since you have shown that , that means

.

Since you know that the two sums on the right are convergent, their sum is also convergent, and anything less than that is also convergent.

Q.E.D.

Re: Show that a series converges absolutely given convergence of two related series

Quote:

Originally Posted by

**tomcruisex** Show that if

and

are convergent then

is absolutely convergent.

Hint: Show |xy| <= 1/2(|x|^2 + |y|^2) for any x,y ε R

I can prove the Hint -

And this is true for all x, y as the square of any real number will be greater than or equal to 0.

But I cant see how to apply this to the original question?

Setting it is easy to demonstrate that the series converges. But is so that the series also converges...

Kind regards

Re: Show that a series converges absolutely given convergence of two related series

Re: Show that a series converges absolutely given convergence of two related series

Thank you for the help guys.

There is one thing I am still not sure about - what is the meaning of ?

Does that mean that the series is made up by squaring each term and adding them together - a0a0 + a1a1 + a2a2 + ... + anan?

Or does it mean that the series you get the sum of series and then square it?

Re: Show that a series converges absolutely given convergence of two related series

Quote:

Originally Posted by

**tomcruisex** Thank you for the help guys.

There is one thing I am still not sure about - what is the meaning of

?

Does that mean that the series is made up by squaring each term and adding them together - a0a0 + a1a1 + a2a2 + ... + anan?

Or does it mean that the series you get the sum of series

and then square it?

The meaning of is [probably...] that You have a series with all positive terms. The reason of that is easily understandable considering the case , where both the series and converge but the series diverges...

Kind regards