Convergence of term-wise product of convergent series
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Let $sum a_n$ and $sum b_n$ be two convergent series. It is easy to prove that their term-wise product $sum a_n b_n$ converges if $a_n,b_n geq 0$, but $sum a_n b_n$ does not necessarily converge otherwise.
My question is, must $sum a_n b_n$ converge if $a_n geq 0$? Having thought about it some, it seems that there should be a counterexample, but I haven't been able to find one.
calculus real-analysis sequences-and-series limits convergence
asked Nov 19 at 23:13
AlephNull
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up vote
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Let $sum a_n$ and $sum b_n$ be two convergent series. It is easy to prove that their term-wise product $sum a_n b_n$ converges if $a_n,b_n geq 0$, but $sum a_n b_n$ does not necessarily converge otherwise.
My question is, must $sum a_n b_n$ converge if $a_n geq 0$? Having thought about it some, it seems that there should be a counterexample, but I haven't been able to find one.
calculus real-analysis sequences-and-series limits convergence
asked Nov 19 at 23:13
AlephNull
224
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Let $sum a_n$ and $sum b_n$ be two convergent series. It is easy to prove that their term-wise product $sum a_n b_n$ converges if $a_n,b_n geq 0$, but $sum a_n b_n$ does not necessarily converge otherwise.
My question is, must $sum a_n b_n$ converge if $a_n geq 0$? Having thought about it some, it seems that there should be a counterexample, but I haven't been able to find one.
calculus real-analysis sequences-and-series limits convergence
asked Nov 19 at 23:13
AlephNull
224
Let $sum a_n$ and $sum b_n$ be two convergent series. It is easy to prove that their term-wise product $sum a_n b_n$ converges if $a_n,b_n geq 0$, but $sum a_n b_n$ does not necessarily converge otherwise.
My question is, must $sum a_n b_n$ converge if $a_n geq 0$? Having thought about it some, it seems that there should be a counterexample, but I haven't been able to find one.
calculus real-analysis sequences-and-series limits convergence
calculus real-analysis sequences-and-series limits convergence
asked Nov 19 at 23:13
AlephNull
224
asked Nov 19 at 23:13
AlephNull
224
asked Nov 19 at 23:13
AlephNull
224
asked Nov 19 at 23:13
AlephNull
224
asked Nov 19 at 23:13
AlephNull
224
224
add a comment |
add a comment |
1 Answer
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$sum |a_n b_n| leq Msum a_n < infty$ where $M=sup_n |b_n|$. Note that $b_n to 0$ so ${b_n}$ is a bounded sequence. Hence $M <infty$ and the series $sum a_n b_n$ is absolutely convergent.
answered Nov 19 at 23:16
Kavi Rama Murthy
46.9k31854
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
accepted
$sum |a_n b_n| leq Msum a_n < infty$ where $M=sup_n |b_n|$. Note that $b_n to 0$ so ${b_n}$ is a bounded sequence. Hence $M <infty$ and the series $sum a_n b_n$ is absolutely convergent.
answered Nov 19 at 23:16
Kavi Rama Murthy
46.9k31854
add a comment |
up vote
1
down vote
accepted
$sum |a_n b_n| leq Msum a_n < infty$ where $M=sup_n |b_n|$. Note that $b_n to 0$ so ${b_n}$ is a bounded sequence. Hence $M <infty$ and the series $sum a_n b_n$ is absolutely convergent.
answered Nov 19 at 23:16
Kavi Rama Murthy
46.9k31854
add a comment |
up vote
1
down vote
accepted
up vote
1
down vote
accepted
$sum |a_n b_n| leq Msum a_n < infty$ where $M=sup_n |b_n|$. Note that $b_n to 0$ so ${b_n}$ is a bounded sequence. Hence $M <infty$ and the series $sum a_n b_n$ is absolutely convergent.
answered Nov 19 at 23:16
Kavi Rama Murthy
46.9k31854
$sum |a_n b_n| leq Msum a_n < infty$ where $M=sup_n |b_n|$. Note that $b_n to 0$ so ${b_n}$ is a bounded sequence. Hence $M <infty$ and the series $sum a_n b_n$ is absolutely convergent.
answered Nov 19 at 23:16
Kavi Rama Murthy
46.9k31854
answered Nov 19 at 23:16
Kavi Rama Murthy
46.9k31854
answered Nov 19 at 23:16
Kavi Rama Murthy
46.9k31854
answered Nov 19 at 23:16
Kavi Rama Murthy
46.9k31854
46.9k31854
add a comment |
add a comment |
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