Absolute convergence, boundedness, and multiplication

[jfy4]

Homework Statement

If the series $\sum_{n=1}^{\infty}x_n$ converges absolutely, and the sequence $(y_n)_n$ is bounded, then the series $\sum_{n=1}^{\infty}x_ny_n$ converges.

Homework Equations

Definitions and theorems relating to series and convergence.

The Attempt at a Solution

If the sequence $y_n$ is bounded, then $|y_n|\leq M$ for all n. Then $\sum_{k=1}^{n}x_ky_k\leq\sum_{k=1}^{n}Mx_k$. But $M\sum_{n=1}^{\infty}x_n$ is convergent which implies that $\sum_{n=1}^{\infty}x_ny_n$ is also by comparison. $\blacksquare$

Now I believe this is the right path, however, I did not use the absolute convergence in this proof, and I didn't really consider if the terms in $y_n$ were negative... I need some help to know how to include these aspects.

 

[Sethric]

$\sum_{k=1}^{n}x_ky_k\leq\sum_{k=1}^{n}Mx_k$

This isn't necessarily true. If both x and y alternate, say y_n = -1ⁿ and x_n = -1ⁿ / n, then it doesn't converge, even though y is bounded and the x-series converges. That is the step in which you need to integrate absolute convergence.

 

[jfy4]

$\sum_{k=1}^{n}x_ky_k\leq\sum_{k=1}^{n}Mx_k$

This isn't necessarily true. If both x and y alternate, say y_n = -1ⁿ and x_n = -1ⁿ / n, then it doesn't converge, even though y is bounded and the x-series converges. That is the step in which you need to integrate absolute convergence.

I agree I need to integrate in absolute convergence, however $x_n=(-1)^n/n$ could not be since it is absolutely convergent, and the alternating harmonic series is not. But I get what you are getting at in that I must find a way to include the absolute convergence for particularly sticky situations.

 

[Sethric]

$$\sum_{n=1}^\infty \frac{(-1)^n}{n}$$ is not absolutely convergent.

and the alternating harmonic series is not

That technically is an alternating harmonic series.

What I am driving at is using absolute values. Can you think of a way to do that step using absolute values that takes advantage of the absolute convergence of x_n?

 

[jfy4]

$$\sum_{n=1}^\infty \frac{(-1)^n}{n}$$ is not absolutely convergent.


That technically is an alternating harmonic series.

What I am driving at is using absolute values. Can you think of a way to do that step using absolute values that takes advantage of the absolute convergence of x_n?

Ah! I was trying to say that it is not absolutely convergent, so $x_n$ could not be what you had in your example. But you posted before I could edit it

I'll try and think a bit more about it but that has been what I have been trying to think of...

 

[Sethric]

Let me try this:

You know the sequence y_n is bounded. What do you know about:

$$\sum_{k=1}^\infty \left| x_k y_k \right|$$

?

 

[jfy4]

Let me try this:

You know the sequence y_n is bounded. What do you know about:

$$\sum_{k=1}^\infty \left| x_k y_k \right|$$

?

Okay, I think I got it:

Given $\sum_{n=1}^{\infty}x_n$ converges absolutely, and $y_n$ is bounded, then $|y_n|\leq M$ for all $n$ for some $M>0$. Consider

$$M\sum_{k=1}^{n}|x_k|\geq\sum_{k=1}^{n}|x_k||y_k|=\sum_{k=1}^{n}|x_ky_k|.$$

But $M\sum_{k=1}^{n}|x_k|$ is convergent which implies that $\sum_{k=1}^{n}|x_ky_k|$ is also convergent. But if the sequence of partial sums converge, so does the series. Then since the series is absolutely convergent, it is convergent. $\blacksquare$