Fdtxjr

How to write a chaotic neutral protagonist and prevent my readers from thinking they are evil?

What does the Digital Threat scope actually do?

Is this Paypal Github SDK reference really a dangerous site?

What will happen if my luggage gets delayed?

Use Mercury as quenching liquid for swords?

ESPP--any reason not to go all in?

Trocar background-image com delay via jQuery

Numerical value of Determinant far from what it is supposed to be

Would those living in a "perfect society" not understand satire

cannot log in to the server after changing SSH port

Having the player face themselves after the mid-game

Did Amazon pay $0 in taxes last year?

Writing text next to a table

Why is it common in European airports not to display the gate for flights until around 45-90 minutes before departure, unlike other places?

What is this tube in a jet engine's air intake?

Create chunks from an array

How do spaceships determine each other's mass in space?

Professor forcing me to attend a conference, I can't afford even with 50% funding

Are small insurances worth it?

What is the purpose of a disclaimer like "this is not legal advice"?

Giving a career talk in my old university, how prominently should I tell students my salary?

Leveling the sagging side of the home

Smooth vector fields on a surface modulo diffeomorphisms

Is it a Cyclops number? "Nobody" knows!

A binomial double sum

Summing a binomial seriesSum of squares of binomial coefficientsFinding the sum of a conditionally convergent double seriesDouble sum and zeta functionEvaluting sum $sum limits_n=0^inftyfracn^kn!$Does this summation (involving binomial) have a closed form? If so, what is it?Finding a closed form for $sum_k=1^infty(-1)^n-1fraczeta(2n+1)2^2n+1$Seeking general formula for Euler Sum $sumlimits _ n=1 ^ infty frac left(H_nright)^pn^q$ with $q$ even and $p$ oddHelp with a nasty sum of binomial coefficientsA Ramanujan sum

1

3

$begingroup$

I was doing some numerical experiment and I found that
$$sum_k=2^inftyfrac1k(k-1)sum_n=k^inftyfrac1binomnk(n+1)=sum_k=2^inftyfrac1(k-1)k^2=2-zeta(2).$$
I wonder if the following variation
$$sum_k=2^inftyfrac1k(k-1)sum_n=k^inftyfrac1binomnk(n-1)$$
has a closed form. Is it related to $zeta(2)$ or $zeta(3)$?

sequences-and-series

share|cite|improve this question

edited Mar 5 at 13:35

Frank S

asked Mar 5 at 12:59

Frank SFrank S

637

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  What have you tried?
  $endgroup$
  – clathratus
  yesterday
  

  
  
  
  

add a comment | 

1

3

$begingroup$

I was doing some numerical experiment and I found that
$$sum_k=2^inftyfrac1k(k-1)sum_n=k^inftyfrac1binomnk(n+1)=sum_k=2^inftyfrac1(k-1)k^2=2-zeta(2).$$
I wonder if the following variation
$$sum_k=2^inftyfrac1k(k-1)sum_n=k^inftyfrac1binomnk(n-1)$$
has a closed form. Is it related to $zeta(2)$ or $zeta(3)$?

sequences-and-series

share|cite|improve this question

edited Mar 5 at 13:35

Frank S

asked Mar 5 at 12:59

Frank SFrank S

637

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  What have you tried?
  $endgroup$
  – clathratus
  yesterday
  

  
  
  
  

add a comment | 

$begingroup$

I was doing some numerical experiment and I found that
$$sum_k=2^inftyfrac1k(k-1)sum_n=k^inftyfrac1binomnk(n+1)=sum_k=2^inftyfrac1(k-1)k^2=2-zeta(2).$$
I wonder if the following variation
$$sum_k=2^inftyfrac1k(k-1)sum_n=k^inftyfrac1binomnk(n-1)$$
has a closed form. Is it related to $zeta(2)$ or $zeta(3)$?

sequences-and-series

share|cite|improve this question

edited Mar 5 at 13:35

Frank S

asked Mar 5 at 12:59

Frank SFrank S

637

$endgroup$

share|cite|improve this question

edited Mar 5 at 13:35

Frank S

asked Mar 5 at 12:59

Frank SFrank S

637

share|cite|improve this question

edited Mar 5 at 13:35

Frank S

asked Mar 5 at 12:59

Frank SFrank S

637

asked Mar 5 at 12:59

Frank SFrank S

637

asked Mar 5 at 12:59

Frank SFrank S

637

1 Answer
1

active

oldest

votes

0

$begingroup$

We have that the inner sum equals
$$
eqalign
& sumlimits_k, le ,n 1 over binomnkleft( n - 1 right) = cr
& = sumlimits_0, le ,n 1 over binomn+kkleft( n + k - 1 right) = cr
& = k!sumlimits_0, le ,n 1 over left( n + k right)^,underline ,k, left( n + k - 1 right) = cr
& = k!sumlimits_0, le ,n 1 over left( n + 1 right)^,overline ,k, left( n + k - 1 right) cr
$$

Therefore
$$
eqalign
& S = sumlimits_2, le ,k 1 over kleft( k - 1 right)sumlimits_k, le ,n 1 over binomnkleft( n - 1 right) = cr
& = sumlimits_2, le ,k k! over kleft( k - 1 right)sumlimits_0, le ,n
1 over left( n + 1 right)^,overline ,k, left( n + k - 1 right) = cr
& = sumlimits_0, le ,k left( k + 2 right)! over left( k + 2 right)left( k + 1 right)sumlimits_0, le ,n
1 over left( n + 1 right)^,overline ,k + 2, left( n + 1 + k right) = cr
& = sumlimits_0, le ,n sumlimits_0, le ,k k! over left( n + 1 right)^,overline ,k + 2, left( n + 1 + k right) = cr
& = sumlimits_0, le ,n sumlimits_0, le ,k k!,left( n + 1 right)^,overline ,k,
over left( n + 1 right)^,overline ,k + 2, left( n + 1 right)^,overline ,k + 1, = cr
& = sumlimits_0, le ,n 1 over left( n + 1 right)^,2 left( n + 2 right)sumlimits_0, le ,k
1^,overline ,k, ,left( n + 1 right)^,overline ,k, over left( n + 3 right)^,overline ,k, left( n + 2 right)^,overline ,k, = cr
& = sumlimits_0, le ,n 1 over left( n + 1 right)^,2 left( n + 2 right)
_3F_,2left( left. matrix 1,;1,n + 1 cr n + 3,n + 2 cr ; right right) cr
$$

Unfortunately, the Hypergeometric does not look to be
of a type expressible in a simpler way.

share|cite|improve this answer

edited yesterday

answered Mar 5 at 16:06

G CabG Cab

19.9k31340

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Are you saying that the given series is divergent? Note that $sum_k=2^inftyfrac1k(k-1)sum_n=k^inftyfrac1binomnk(n-1)leq sum_k=2^inftyfrac1k(k-1)sum_n=k^inftyfrac1binomnk=sum_k=2^inftyfrac1(k-1)^2=zeta(2)$.
  $endgroup$
  – Robert Z
  Mar 5 at 16:22
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @RobertZ: ehps, probably I made some error: will verify, thanks for signalling
  $endgroup$
  – G Cab
  Mar 5 at 17:16
  

  
  
  
  

add a comment | 

Your Answer

StackExchange.ifUsing("editor", function ()
return StackExchange.using("mathjaxEditing", function ()
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix)
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
);
);
, "mathjax-editing");

StackExchange.ready(function()
var channelOptions =
tags: "".split(" "),
id: "69"
;
initTagRenderer("".split(" "), "".split(" "), channelOptions);

StackExchange.using("externalEditor", function()
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled)
StackExchange.using("snippets", function()
createEditor();
);

else
createEditor();

);

function createEditor()
StackExchange.prepareEditor(
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader:
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
,
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
);



);

draft saved

draft discarded

Sign up or log in

StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);

Sign up using Google

Sign up using Facebook

Sign up using Email and Password

Post as a guest

Name

Email

Required, but never shown

StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3136181%2fa-binomial-double-sum%23new-answer', 'question_page');

);

Post as a guest

Name

Email

Required, but never shown

0

$begingroup$

We have that the inner sum equals
$$
eqalign
& sumlimits_k, le ,n 1 over binomnkleft( n - 1 right) = cr
& = sumlimits_0, le ,n 1 over binomn+kkleft( n + k - 1 right) = cr
& = k!sumlimits_0, le ,n 1 over left( n + k right)^,underline ,k, left( n + k - 1 right) = cr
& = k!sumlimits_0, le ,n 1 over left( n + 1 right)^,overline ,k, left( n + k - 1 right) cr
$$

Therefore
$$
eqalign
& S = sumlimits_2, le ,k 1 over kleft( k - 1 right)sumlimits_k, le ,n 1 over binomnkleft( n - 1 right) = cr
& = sumlimits_2, le ,k k! over kleft( k - 1 right)sumlimits_0, le ,n
1 over left( n + 1 right)^,overline ,k, left( n + k - 1 right) = cr
& = sumlimits_0, le ,k left( k + 2 right)! over left( k + 2 right)left( k + 1 right)sumlimits_0, le ,n
1 over left( n + 1 right)^,overline ,k + 2, left( n + 1 + k right) = cr
& = sumlimits_0, le ,n sumlimits_0, le ,k k! over left( n + 1 right)^,overline ,k + 2, left( n + 1 + k right) = cr
& = sumlimits_0, le ,n sumlimits_0, le ,k k!,left( n + 1 right)^,overline ,k,
over left( n + 1 right)^,overline ,k + 2, left( n + 1 right)^,overline ,k + 1, = cr
& = sumlimits_0, le ,n 1 over left( n + 1 right)^,2 left( n + 2 right)sumlimits_0, le ,k
1^,overline ,k, ,left( n + 1 right)^,overline ,k, over left( n + 3 right)^,overline ,k, left( n + 2 right)^,overline ,k, = cr
& = sumlimits_0, le ,n 1 over left( n + 1 right)^,2 left( n + 2 right)
_3F_,2left( left. matrix 1,;1,n + 1 cr n + 3,n + 2 cr ; right right) cr
$$

Unfortunately, the Hypergeometric does not look to be
of a type expressible in a simpler way.

share|cite|improve this answer

edited yesterday

answered Mar 5 at 16:06

G CabG Cab

19.9k31340

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Are you saying that the given series is divergent? Note that $sum_k=2^inftyfrac1k(k-1)sum_n=k^inftyfrac1binomnk(n-1)leq sum_k=2^inftyfrac1k(k-1)sum_n=k^inftyfrac1binomnk=sum_k=2^inftyfrac1(k-1)^2=zeta(2)$.
  $endgroup$
  – Robert Z
  Mar 5 at 16:22
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @RobertZ: ehps, probably I made some error: will verify, thanks for signalling
  $endgroup$
  – G Cab
  Mar 5 at 17:16
  

  
  
  
  

add a comment | 

0

$begingroup$

We have that the inner sum equals
$$
eqalign
& sumlimits_k, le ,n 1 over binomnkleft( n - 1 right) = cr
& = sumlimits_0, le ,n 1 over binomn+kkleft( n + k - 1 right) = cr
& = k!sumlimits_0, le ,n 1 over left( n + k right)^,underline ,k, left( n + k - 1 right) = cr
& = k!sumlimits_0, le ,n 1 over left( n + 1 right)^,overline ,k, left( n + k - 1 right) cr
$$

Therefore
$$
eqalign
& S = sumlimits_2, le ,k 1 over kleft( k - 1 right)sumlimits_k, le ,n 1 over binomnkleft( n - 1 right) = cr
& = sumlimits_2, le ,k k! over kleft( k - 1 right)sumlimits_0, le ,n
1 over left( n + 1 right)^,overline ,k, left( n + k - 1 right) = cr
& = sumlimits_0, le ,k left( k + 2 right)! over left( k + 2 right)left( k + 1 right)sumlimits_0, le ,n
1 over left( n + 1 right)^,overline ,k + 2, left( n + 1 + k right) = cr
& = sumlimits_0, le ,n sumlimits_0, le ,k k! over left( n + 1 right)^,overline ,k + 2, left( n + 1 + k right) = cr
& = sumlimits_0, le ,n sumlimits_0, le ,k k!,left( n + 1 right)^,overline ,k,
over left( n + 1 right)^,overline ,k + 2, left( n + 1 right)^,overline ,k + 1, = cr
& = sumlimits_0, le ,n 1 over left( n + 1 right)^,2 left( n + 2 right)sumlimits_0, le ,k
1^,overline ,k, ,left( n + 1 right)^,overline ,k, over left( n + 3 right)^,overline ,k, left( n + 2 right)^,overline ,k, = cr
& = sumlimits_0, le ,n 1 over left( n + 1 right)^,2 left( n + 2 right)
_3F_,2left( left. matrix 1,;1,n + 1 cr n + 3,n + 2 cr ; right right) cr
$$

Unfortunately, the Hypergeometric does not look to be
of a type expressible in a simpler way.

share|cite|improve this answer

edited yesterday

answered Mar 5 at 16:06

G CabG Cab

19.9k31340

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Are you saying that the given series is divergent? Note that $sum_k=2^inftyfrac1k(k-1)sum_n=k^inftyfrac1binomnk(n-1)leq sum_k=2^inftyfrac1k(k-1)sum_n=k^inftyfrac1binomnk=sum_k=2^inftyfrac1(k-1)^2=zeta(2)$.
  $endgroup$
  – Robert Z
  Mar 5 at 16:22
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @RobertZ: ehps, probably I made some error: will verify, thanks for signalling
  $endgroup$
  – G Cab
  Mar 5 at 17:16
  

  
  
  
  

add a comment | 

$begingroup$

We have that the inner sum equals
$$
eqalign
& sumlimits_k, le ,n 1 over binomnkleft( n - 1 right) = cr
& = sumlimits_0, le ,n 1 over binomn+kkleft( n + k - 1 right) = cr
& = k!sumlimits_0, le ,n 1 over left( n + k right)^,underline ,k, left( n + k - 1 right) = cr
& = k!sumlimits_0, le ,n 1 over left( n + 1 right)^,overline ,k, left( n + k - 1 right) cr
$$

Therefore
$$
eqalign
& S = sumlimits_2, le ,k 1 over kleft( k - 1 right)sumlimits_k, le ,n 1 over binomnkleft( n - 1 right) = cr
& = sumlimits_2, le ,k k! over kleft( k - 1 right)sumlimits_0, le ,n
1 over left( n + 1 right)^,overline ,k, left( n + k - 1 right) = cr
& = sumlimits_0, le ,k left( k + 2 right)! over left( k + 2 right)left( k + 1 right)sumlimits_0, le ,n
1 over left( n + 1 right)^,overline ,k + 2, left( n + 1 + k right) = cr
& = sumlimits_0, le ,n sumlimits_0, le ,k k! over left( n + 1 right)^,overline ,k + 2, left( n + 1 + k right) = cr
& = sumlimits_0, le ,n sumlimits_0, le ,k k!,left( n + 1 right)^,overline ,k,
over left( n + 1 right)^,overline ,k + 2, left( n + 1 right)^,overline ,k + 1, = cr
& = sumlimits_0, le ,n 1 over left( n + 1 right)^,2 left( n + 2 right)sumlimits_0, le ,k
1^,overline ,k, ,left( n + 1 right)^,overline ,k, over left( n + 3 right)^,overline ,k, left( n + 2 right)^,overline ,k, = cr
& = sumlimits_0, le ,n 1 over left( n + 1 right)^,2 left( n + 2 right)
_3F_,2left( left. matrix 1,;1,n + 1 cr n + 3,n + 2 cr ; right right) cr
$$

Unfortunately, the Hypergeometric does not look to be
of a type expressible in a simpler way.

share|cite|improve this answer

edited yesterday

answered Mar 5 at 16:06

G CabG Cab

19.9k31340

$endgroup$

share|cite|improve this answer

edited yesterday

answered Mar 5 at 16:06

G CabG Cab

19.9k31340

answered Mar 5 at 16:06

G CabG Cab

19.9k31340

answered Mar 5 at 16:06

G CabG Cab

19.9k31340