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Question about sums and double sumsUniform convergence of the series on unbounded domaininterchanging sum and lim with uniformly convergence seriesShow that the sequence of functions ${f_n}(x)=frac{x}{1+nx^2}$ converges uniformlyConvergence of series implies convergence of related seriesDisproving uniform convergence of $sum_{n=1}^infty frac2{pi} frac1n Big (1-cosBig(nfrac{pi}2Big) Big)sin(nx)$Find a sequence $f_n$ so that $int_0 ^1 |f_n(x)| = 2$ and $lim_{n to infty} f_n(x) = 1$.$n$-th partial sum and convergence $sum_{k=1}^{infty}frac{1}{k(k+2)}$Sequence $(f_n) to f_n$ on $S subseteq mathbb{R}$ converges uniformly iff $lim_{n to infty} sup {f(x)-f_n(x)| : xin S} = 0$Show $ langle f(t), f'(t) rangle = 0$ for all $t in mathbb{R}$.

0

$begingroup$

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 $$sum_{n = 1}^infty left (sum_{k=1}^infty f_n(k)right) = infty.$$ Prove or disprove that $$sum_{k = 1}^infty left (sum_{n=1}^infty f_n(k)right) = infty.$$

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

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edited yesterday

Xander Henderson

14.8k103555

asked Jan 21 at 0:37

user439126user439126

1586

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put on hold as off-topic by Carl Mummert, RRL, Xander Henderson, Saad, GNUSupporter 8964民主女神 地下教會 18 hours ago

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – Carl Mummert, RRL, Xander Henderson, Saad, GNUSupporter 8964民主女神 地下教會

If this question can be reworded to fit the rules in the help center, please edit the question.

• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  $begingroup$
  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 | 

0

$begingroup$

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 $$sum_{n = 1}^infty left (sum_{k=1}^infty f_n(k)right) = infty.$$ Prove or disprove that $$sum_{k = 1}^infty left (sum_{n=1}^infty f_n(k)right) = infty.$$

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

share|cite|improve this question

edited yesterday

Xander Henderson

14.8k103555

asked Jan 21 at 0:37

user439126user439126

1586

$endgroup$

put on hold as off-topic by Carl Mummert, RRL, Xander Henderson, Saad, GNUSupporter 8964民主女神 地下教會 18 hours ago

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – Carl Mummert, RRL, Xander Henderson, Saad, GNUSupporter 8964民主女神 地下教會

If this question can be reworded to fit the rules in the help center, please edit the question.

• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  $begingroup$
  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 | 

$begingroup$

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 $$sum_{n = 1}^infty left (sum_{k=1}^infty f_n(k)right) = infty.$$ Prove or disprove that $$sum_{k = 1}^infty left (sum_{n=1}^infty f_n(k)right) = infty.$$

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

share|cite|improve this question

edited yesterday

Xander Henderson

14.8k103555

asked Jan 21 at 0:37

user439126user439126

1586

$endgroup$

share|cite|improve this question

edited yesterday

Xander Henderson

14.8k103555

asked Jan 21 at 0:37

user439126user439126

1586

share|cite|improve this question

edited yesterday

Xander Henderson

14.8k103555

asked Jan 21 at 0:37

user439126user439126

1586

edited yesterday

Xander Henderson

14.8k103555

edited yesterday

Xander Henderson

14.8k103555

asked Jan 21 at 0:37

user439126user439126

1586

asked Jan 21 at 0:37

user439126user439126

1586

1 Answer
1

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oldest

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1

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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 :
$$ sup_{K, N in mathbf {N}} sum_{n=0}^N sum_{k=0}^K f_n (k) $$

share|cite|improve this answer

answered Jan 21 at 0:48

DLeMeurDLeMeur

3248

$endgroup$

add a comment | 

1

$begingroup$

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 :
$$ sup_{K, N in mathbf {N}} sum_{n=0}^N sum_{k=0}^K f_n (k) $$

share|cite|improve this answer

answered Jan 21 at 0:48

DLeMeurDLeMeur

3248

$endgroup$

add a comment | 

1

$begingroup$

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 :
$$ sup_{K, N in mathbf {N}} sum_{n=0}^N sum_{k=0}^K f_n (k) $$

share|cite|improve this answer

answered Jan 21 at 0:48

DLeMeurDLeMeur

3248

$endgroup$

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 :
$$ sup_{K, N in mathbf {N}} sum_{n=0}^N sum_{k=0}^K f_n (k) $$

share|cite|improve this answer

answered Jan 21 at 0:48

DLeMeurDLeMeur

3248

$endgroup$

share|cite|improve this answer

answered Jan 21 at 0:48

DLeMeurDLeMeur

3248

answered Jan 21 at 0:48

DLeMeurDLeMeur

3248

answered Jan 21 at 0:48

DLeMeurDLeMeur

3248