(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

I'm been trying to wrap my head around this one for a couple of days. I have a problem that states the following.

(All sums go from n to [tex]\infty[/tex]" Suppose we know that [tex]\sum[/tex]A_{n }and [tex]\sum[/tex] B_{ n }are both convergent series each with positive terms.

Show that the series [tex]\sum[/tex]A_{n }B_{n}must be convergent. I thought that if I took the first three terms of the series and multiply them together I could argue that since the terms alone are convergent then the product must be as well. However I'm not sure of that is the correct way to go about it.

Maybe I should take the derivative of each product and then make that argument. Anyone

3. The attempt at a solution

Here's what I wrote down while trying to solve this problem.

f(x) = [tex]\sum[/tex]A_{n}

g(x) = [tex]\sum[/tex]B_{n}

Known:

[tex]\sum[/tex]A_{n}for all values

[tex]\sum[/tex]B_{n}for all values

f(0) + f(1) + f(2) = A_{0}+ A_{1}+ A_{2}

g(0) + g(1) + g(2) = B_{0}+ B_{1}+ B_{2}

f(x) * g(x) = [tex]\sum[/tex]A_{n}B_{n}

= A_{0}+ A_{1}+ A_{2}(B_{0}+ B_{1}+ B_{2}

)

[From here I foil the expression, which is a nightmare type unless you type at 120 wpm. So I'm going to skip that part.]

Since all values of [tex]\sum[/tex]A_{n}and [tex]\sum[/tex]B_{n}convergent it follows that [tex]\sum[/tex]A_{n}B_{n}

is convergent as well.

Since I know that's not going to fly given there isn't all real proof shown, what else could I do? Do I take a derivative as mentioned or can I express the two sums in their elementary state?

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# Homework Help: Proving convergent for the products of two series.

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