Sum of series $sumlimits_{k=0}^infty frac{a^{k^2}}{k!}$
$begingroup$
I am looking at this sum:
$displaystylesum_{k=0}^infty frac{a^{(k^2)}}{k!}$ for some $a>0$.
sequences-and-series power-series
edited Dec 4 '18 at 20:47
Did
247k23223459
asked Dec 4 '18 at 19:49
E OnaranE Onaran
62
$endgroup$
add a comment |
$begingroup$
I am looking at this sum:
$displaystylesum_{k=0}^infty frac{a^{(k^2)}}{k!}$ for some $a>0$.
sequences-and-series power-series
edited Dec 4 '18 at 20:47
Did
247k23223459
asked Dec 4 '18 at 19:49
E OnaranE Onaran
62
$endgroup$
4
$begingroup$
Hi and welcome to the Math.SE. Why this problem is interesting for you? And what did you tried in order to solve it? I you want to get help from other members, you should try to add as more context as possible to your question, in order to help them help you.
$endgroup$
– Daniele Tampieri
Dec 4 '18 at 19:52
add a comment |
$begingroup$
I am looking at this sum:
$displaystylesum_{k=0}^infty frac{a^{(k^2)}}{k!}$ for some $a>0$.
sequences-and-series power-series
edited Dec 4 '18 at 20:47
Did
247k23223459
asked Dec 4 '18 at 19:49
E OnaranE Onaran
62
$endgroup$
I am looking at this sum:
$displaystylesum_{k=0}^infty frac{a^{(k^2)}}{k!}$ for some $a>0$.
sequences-and-series power-series
sequences-and-series power-series
edited Dec 4 '18 at 20:47
Did
247k23223459
asked Dec 4 '18 at 19:49
E OnaranE Onaran
62
edited Dec 4 '18 at 20:47
Did
247k23223459
asked Dec 4 '18 at 19:49
E OnaranE Onaran
62
edited Dec 4 '18 at 20:47
Did
247k23223459
edited Dec 4 '18 at 20:47
Did
247k23223459
edited Dec 4 '18 at 20:47
Did
247k23223459
247k23223459
asked Dec 4 '18 at 19:49
E OnaranE Onaran
62
asked Dec 4 '18 at 19:49
E OnaranE Onaran
62
asked Dec 4 '18 at 19:49
E OnaranE Onaran
62
62
4
$begingroup$
Hi and welcome to the Math.SE. Why this problem is interesting for you? And what did you tried in order to solve it? I you want to get help from other members, you should try to add as more context as possible to your question, in order to help them help you.
$endgroup$
– Daniele Tampieri
Dec 4 '18 at 19:52
add a comment |
4
$begingroup$
Hi and welcome to the Math.SE. Why this problem is interesting for you? And what did you tried in order to solve it? I you want to get help from other members, you should try to add as more context as possible to your question, in order to help them help you.
$endgroup$
– Daniele Tampieri
Dec 4 '18 at 19:52
4
4
$begingroup$
Hi and welcome to the Math.SE. Why this problem is interesting for you? And what did you tried in order to solve it? I you want to get help from other members, you should try to add as more context as possible to your question, in order to help them help you.
$endgroup$
– Daniele Tampieri
Dec 4 '18 at 19:52
$begingroup$
Hi and welcome to the Math.SE. Why this problem is interesting for you? And what did you tried in order to solve it? I you want to get help from other members, you should try to add as more context as possible to your question, in order to help them help you.
$endgroup$
– Daniele Tampieri
Dec 4 '18 at 19:52
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
If I get to choose $a$ I choose $1$ and the sum is $e$.
For $a gt 1$ the sum diverges.
For $0 lt a lt 1$ it will converge quickly but I suspect has no easy answer. Alpha says that for $a=frac 12$ the sum is about $0.531576$ but does not give a closed form.
answered Dec 4 '18 at 20:52
Ross MillikanRoss Millikan
294k23198371
$endgroup$
$begingroup$
This is a special function, theta or something.
$endgroup$
– Did
Dec 4 '18 at 23:57
add a comment |
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
If I get to choose $a$ I choose $1$ and the sum is $e$.
For $a gt 1$ the sum diverges.
For $0 lt a lt 1$ it will converge quickly but I suspect has no easy answer. Alpha says that for $a=frac 12$ the sum is about $0.531576$ but does not give a closed form.
answered Dec 4 '18 at 20:52
Ross MillikanRoss Millikan
294k23198371
$endgroup$
$begingroup$
This is a special function, theta or something.
$endgroup$
– Did
Dec 4 '18 at 23:57
add a comment |
$begingroup$
If I get to choose $a$ I choose $1$ and the sum is $e$.
For $a gt 1$ the sum diverges.
For $0 lt a lt 1$ it will converge quickly but I suspect has no easy answer. Alpha says that for $a=frac 12$ the sum is about $0.531576$ but does not give a closed form.
answered Dec 4 '18 at 20:52
Ross MillikanRoss Millikan
294k23198371
$endgroup$
$begingroup$
This is a special function, theta or something.
$endgroup$
– Did
Dec 4 '18 at 23:57
add a comment |
$begingroup$
If I get to choose $a$ I choose $1$ and the sum is $e$.
For $a gt 1$ the sum diverges.
For $0 lt a lt 1$ it will converge quickly but I suspect has no easy answer. Alpha says that for $a=frac 12$ the sum is about $0.531576$ but does not give a closed form.
answered Dec 4 '18 at 20:52
Ross MillikanRoss Millikan
294k23198371
$endgroup$
If I get to choose $a$ I choose $1$ and the sum is $e$.
For $a gt 1$ the sum diverges.
For $0 lt a lt 1$ it will converge quickly but I suspect has no easy answer. Alpha says that for $a=frac 12$ the sum is about $0.531576$ but does not give a closed form.
answered Dec 4 '18 at 20:52
Ross MillikanRoss Millikan
294k23198371
answered Dec 4 '18 at 20:52
Ross MillikanRoss Millikan
294k23198371
answered Dec 4 '18 at 20:52
Ross MillikanRoss Millikan
294k23198371
answered Dec 4 '18 at 20:52
Ross MillikanRoss Millikan
294k23198371
294k23198371
$begingroup$
This is a special function, theta or something.
$endgroup$
– Did
Dec 4 '18 at 23:57
add a comment |
$begingroup$
This is a special function, theta or something.
$endgroup$
– Did
Dec 4 '18 at 23:57
$begingroup$
This is a special function, theta or something.
$endgroup$
– Did
Dec 4 '18 at 23:57
$begingroup$
This is a special function, theta or something.
$endgroup$
– Did
Dec 4 '18 at 23:57
add a comment |
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$begingroup$
Hi and welcome to the Math.SE. Why this problem is interesting for you? And what did you tried in order to solve it? I you want to get help from other members, you should try to add as more context as possible to your question, in order to help them help you.
$endgroup$
– Daniele Tampieri
Dec 4 '18 at 19:52