Real Analysis

[math25]

Hi,

Is it possible to prove that if the sum ai and sum bi are convergent series then sum aibi converges? If so, can someone please help me with this problem.

thank you

 

[pka]

Is it possible to prove that if the sum ai and sum bi are convergent series then sum aibi converges? If so, can someone please help me with this problem.

What do you think about these two series:
\(\displaystyle \sum\limits_{k = 1}^\infty {\frac{{{{\left( { - 1} \right)}^k}}}{{\sqrt k }}} \;\& \quad \sum\limits_{k = 1}^\infty {\frac{{{{\left( { - 1} \right)}^{k + 1}}}}{{\sqrt k }}} \)

 

[math25]

the product of those two series doesn't converge?

Does it change anything if we know that the sum ai and sum bi are convergent series with non-negative terms? thanks

 

[pka]

the product of those two series doesn't converge?
Does it change anything if we know that the sum ai and sum bi are convergent series with non-negative terms?

Yes it makes a real difference.
If \(\displaystyle \sum {{a_n}}\) converges then so does \(\displaystyle \sum {{(a_n)^2}}\)

Now consider \(\displaystyle (a_n+b_n)^2\).

 

[math25]

thank you so much for all your help.

is it true that product of those two series doesn't converge?

 

[math25]

What do you think about these two series:
\(\displaystyle \sum\limits_{k = 1}^\infty {\frac{{{{\left( { - 1} \right)}^k}}}{{\sqrt k }}} \;\& \quad \sum\limits_{k = 1}^\infty {\frac{{{{\left( { - 1} \right)}^{k + 1}}}}{{\sqrt k }}} \)

I meant the product of those two series.

 

[pka]

I meant the product of those two series.

That is for non-negative series.
It each of two non-negative series converges then so does the series \(\displaystyle \sum {{a_n}} {b_n}\).