Series convergence- why is my proof incorrect?

[JThompson]

Homework Statement

Prove that if $\sum{|a_{n}|}$ converges and $(b_{n})$ is a bounded sequence, then $\sum a_{n}b_{n}$ converges.

Homework Equations

Comparison Test part (i): Let $\sum a_{n}$ be a series where $a_{n}\geq 0$ for all n. If $\sum a_{n}$ converges and $|b_{n}|\leq a_{n}$ for all n, then $\sum b_{n}$ converges.

or Cauchy criterion

The Attempt at a Solution

Upon seeing this problem, I immediately thought of a proof using the Comparison Test which seemed easier that using the Cauchy criterion.
Proof:

Since $(b_{n})$ is bounded, $\exists M\in\mathbb{R}$ with $|b_{n}|\leq M$ for all n. Then $|a_{n}b_{n}|\leq M|a_{n}|$.
Since $\sum |a_{n}|$ converges, $\sum M|a_{n}|$ converges (we proved this previously).
Since $|a_{n}b_{n}|\leq M|a_{n}|$ and $\sum M|a_{n}|$ converges, $\sum a_{n}b_{n}$ converges by the Comparison Test.

There were no marks indicating which part was incorrect, but I received half credit for the problem and my professor's comment was, "$\sum |a_{n}b_{n}|$ converges, use Cauchy criterion." Where does this proof run afoul?

 

[vela]

As your professor's note indicates, you showed that Σ |a_nb_n| converges. You didn't show Σ a_nb_n converges yet.

 

[JThompson]

But according to the Comparison Test

Comparison Test part (i): Let $\sum a_{n}$ be a series where $a_{n}\geq 0$ for all n. If $\sum a_{n}$ converges and $|b_{n}|\leq a_{n}$ for all n, then $\sum b_{n}$ converges.

$M|a_{n}|\geq 0$ is obvious since $M\geq 0$. I proved that $\sum M|a_{n}|$ converges and that $|a_{n}b_{n}|\leq M|a_{n}|$. It seems to me by the Comparison Test above that $\sum a_{n}b_{n}$. The final series in the Comparison Test is not an absolute value, so why can I not do the same in this problem? How am I applying the Comparison Test incorrectly?

 

[vela]

Yeah, you're right. Perhaps someone else can spot what we're both missing, or maybe your professor is wrong. You could go ask for clarification.

 

[JThompson]

I'll ask him if no one responds- it's not urgent- but I have an easier time articulating questions online (because I'm shy), and written responses are easier to understand than verbal responses.

 

[statdad]

As your professor's note indicates, you showed that Σ |a_nb_n| converges. You didn't show Σ a_nb_n converges yet.

Yes he did.
If

$$\sum |a_nb_n|$$

then

$$\sum a_n b_n$$

converges (absolute convergence)

 

[vela]

Yes he did.
If

$$\sum |a_nb_n|$$

then

$$\sum a_n b_n$$

converges (absolute convergence)

D'oh! I hate it when I miss obvious stuff like that.

I've asked for others to take a look at this thread to see why the professor didn't like the proof.

 

[D H]

Did the problem statement tell you to use the Cauchy criteria?

 

[JThompson]

The only part of the problem statement that I did not post above was the book's hint.

Hint: Use Theorem 14.4

Theorem 14.4 states that a series converges iff it satisfies the Cauchy Criterion.
A hint is a suggestion, not a requirement. At least, I assumed as much.

 

[jbunniii]

I see nothing wrong with your proof. Using the Cauchy criterion would only obfuscate it, in my opinion.