Double sums and convergence to $infty$.
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I am having trouble proving/disproving the following:
Let ${f_n}_{n=1}^infty$ be a sequence with $f_n:mathbb{N} rightarrow mathbb{R}^+$.
Suppose $sum_{n = 1}^infty big (sum_{k=1}^infty f_n(k)) = infty$. Prove or disprove that $sum_{k = 1}^infty big (sum_{n=1}^infty f_n(k)) = infty$.
I suspect it is true, and have tried using the definition of a series diverging to $+ infty$, but I don't know how to handle the inner sum. Any help would be appreciated.
real-analysis
asked Jan 21 at 0:37
user439126user439126
1196
$endgroup$
add a comment |
$begingroup$
I am having trouble proving/disproving the following:
Let ${f_n}_{n=1}^infty$ be a sequence with $f_n:mathbb{N} rightarrow mathbb{R}^+$.
Suppose $sum_{n = 1}^infty big (sum_{k=1}^infty f_n(k)) = infty$. Prove or disprove that $sum_{k = 1}^infty big (sum_{n=1}^infty f_n(k)) = infty$.
I suspect it is true, and have tried using the definition of a series diverging to $+ infty$, but I don't know how to handle the inner sum. Any help would be appreciated.
real-analysis
asked Jan 21 at 0:37
user439126user439126
1196
$endgroup$
1
$begingroup$
Try proving the contrapositive (that is, if one is finite, then they’re both finite [and have the same value]). Hint: all of the terms are positive.
$endgroup$
– Clayton
Jan 21 at 0:44
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With the contrapositive, I don't have to worry about divergence (or oscillations) since all the terms are positive , right?
$endgroup$
– user439126
Jan 21 at 0:49
$begingroup$
Correct. ${}{}$
$endgroup$
– Clayton
Jan 21 at 0:55
$begingroup$
Great, thanks a bunch.
$endgroup$
– user439126
Jan 21 at 0:59
add a comment |
$begingroup$
I am having trouble proving/disproving the following:
Let ${f_n}_{n=1}^infty$ be a sequence with $f_n:mathbb{N} rightarrow mathbb{R}^+$.
Suppose $sum_{n = 1}^infty big (sum_{k=1}^infty f_n(k)) = infty$. Prove or disprove that $sum_{k = 1}^infty big (sum_{n=1}^infty f_n(k)) = infty$.
I suspect it is true, and have tried using the definition of a series diverging to $+ infty$, but I don't know how to handle the inner sum. Any help would be appreciated.
real-analysis
asked Jan 21 at 0:37
user439126user439126
1196
$endgroup$
I am having trouble proving/disproving the following:
Let ${f_n}_{n=1}^infty$ be a sequence with $f_n:mathbb{N} rightarrow mathbb{R}^+$.
Suppose $sum_{n = 1}^infty big (sum_{k=1}^infty f_n(k)) = infty$. Prove or disprove that $sum_{k = 1}^infty big (sum_{n=1}^infty f_n(k)) = infty$.
I suspect it is true, and have tried using the definition of a series diverging to $+ infty$, but I don't know how to handle the inner sum. Any help would be appreciated.
real-analysis
real-analysis
asked Jan 21 at 0:37
user439126user439126
1196
asked Jan 21 at 0:37
user439126user439126
1196
asked Jan 21 at 0:37
user439126user439126
1196
asked Jan 21 at 0:37
user439126user439126
1196
asked Jan 21 at 0:37
user439126user439126
1196
1196
1
$begingroup$
Try proving the contrapositive (that is, if one is finite, then they’re both finite [and have the same value]). Hint: all of the terms are positive.
$endgroup$
– Clayton
Jan 21 at 0:44
$begingroup$
With the contrapositive, I don't have to worry about divergence (or oscillations) since all the terms are positive , right?
$endgroup$
– user439126
Jan 21 at 0:49
$begingroup$
Correct. ${}{}$
$endgroup$
– Clayton
Jan 21 at 0:55
$begingroup$
Great, thanks a bunch.
$endgroup$
– user439126
Jan 21 at 0:59
add a comment |
1
$begingroup$
Try proving the contrapositive (that is, if one is finite, then they’re both finite [and have the same value]). Hint: all of the terms are positive.
$endgroup$
– Clayton
Jan 21 at 0:44
$begingroup$
With the contrapositive, I don't have to worry about divergence (or oscillations) since all the terms are positive , right?
$endgroup$
– user439126
Jan 21 at 0:49
$begingroup$
Correct. ${}{}$
$endgroup$
– Clayton
Jan 21 at 0:55
$begingroup$
Great, thanks a bunch.
$endgroup$
– user439126
Jan 21 at 0:59
1
1
$begingroup$
Try proving the contrapositive (that is, if one is finite, then they’re both finite [and have the same value]). Hint: all of the terms are positive.
$endgroup$
– Clayton
Jan 21 at 0:44
$begingroup$
Try proving the contrapositive (that is, if one is finite, then they’re both finite [and have the same value]). Hint: all of the terms are positive.
$endgroup$
– Clayton
Jan 21 at 0:44
$begingroup$
With the contrapositive, I don't have to worry about divergence (or oscillations) since all the terms are positive , right?
$endgroup$
– user439126
Jan 21 at 0:49
$begingroup$
With the contrapositive, I don't have to worry about divergence (or oscillations) since all the terms are positive , right?
$endgroup$
– user439126
Jan 21 at 0:49
$begingroup$
Correct. ${}{}$
$endgroup$
– Clayton
Jan 21 at 0:55
$begingroup$
Correct. ${}{}$
$endgroup$
– Clayton
Jan 21 at 0:55
$begingroup$
Great, thanks a bunch.
$endgroup$
– user439126
Jan 21 at 0:59
$begingroup$
Great, thanks a bunch.
$endgroup$
– user439126
Jan 21 at 0:59
add a comment |
1 Answer
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Hint : Yes it is true, because the terms on the sum are positive. So you can show that both terms are equal to the supremum of the finite sums :
$$ sup_{K, N in mathbf {N}} sum_{n=0}^N sum_{k=0}^K f_n (k) $$
answered Jan 21 at 0:48
DLeMeurDLeMeur
3148
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add a comment |
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1 Answer
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1 Answer
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$begingroup$
Hint : Yes it is true, because the terms on the sum are positive. So you can show that both terms are equal to the supremum of the finite sums :
$$ sup_{K, N in mathbf {N}} sum_{n=0}^N sum_{k=0}^K f_n (k) $$
answered Jan 21 at 0:48
DLeMeurDLeMeur
3148
$endgroup$
add a comment |
$begingroup$
Hint : Yes it is true, because the terms on the sum are positive. So you can show that both terms are equal to the supremum of the finite sums :
$$ sup_{K, N in mathbf {N}} sum_{n=0}^N sum_{k=0}^K f_n (k) $$
answered Jan 21 at 0:48
DLeMeurDLeMeur
3148
$endgroup$
add a comment |
$begingroup$
Hint : Yes it is true, because the terms on the sum are positive. So you can show that both terms are equal to the supremum of the finite sums :
$$ sup_{K, N in mathbf {N}} sum_{n=0}^N sum_{k=0}^K f_n (k) $$
answered Jan 21 at 0:48
DLeMeurDLeMeur
3148
$endgroup$
Hint : Yes it is true, because the terms on the sum are positive. So you can show that both terms are equal to the supremum of the finite sums :
$$ sup_{K, N in mathbf {N}} sum_{n=0}^N sum_{k=0}^K f_n (k) $$
answered Jan 21 at 0:48
DLeMeurDLeMeur
3148
answered Jan 21 at 0:48
DLeMeurDLeMeur
3148
answered Jan 21 at 0:48
DLeMeurDLeMeur
3148
answered Jan 21 at 0:48
DLeMeurDLeMeur
3148
3148
add a comment |
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$begingroup$
Try proving the contrapositive (that is, if one is finite, then they’re both finite [and have the same value]). Hint: all of the terms are positive.
$endgroup$
– Clayton
Jan 21 at 0:44
$begingroup$
With the contrapositive, I don't have to worry about divergence (or oscillations) since all the terms are positive , right?
$endgroup$
– user439126
Jan 21 at 0:49
$begingroup$
Correct. ${}{}$
$endgroup$
– Clayton
Jan 21 at 0:55
$begingroup$
Great, thanks a bunch.
$endgroup$
– user439126
Jan 21 at 0:59