Evaluation of :$sum_{ngeq 2}frac1nGamma(frac1n)^{zeta{(frac1n)}}$ [closed]
I want to know more about behavior of both Gamma function and zeta function writing them as a power in the form of harmonic series which i got the below form
$$sum_{ngeq 2}dfrac1nleft(Gammaleft(frac1nright)^{zeta{left(frac1nright)}}right)$$
which seems converge as shown by Wolfram alpha it's seems converge approximately to $1.6$ But no way to get it's closed form since it's complicated , Now i want the exact value of that sum and it's closed value form if it exists however my narrow idea and attempts are fail to get it .
Note: The Motivation of this question is to know bounds of gamma function power zeta function for $n$ large enough
sequences-and-series gamma-function riemann-zeta
asked Nov 27 '18 at 20:15
zeraoulia rafik
2,38411029
closed as off-topic by user21820, Did, metamorphy, José Carlos Santos, Cesareo Dec 20 '18 at 12:10
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." – Did, metamorphy, José Carlos Santos, Cesareo
If this question can be reworded to fit the rules in the help center, please edit the question.
add a comment |
I want to know more about behavior of both Gamma function and zeta function writing them as a power in the form of harmonic series which i got the below form
$$sum_{ngeq 2}dfrac1nleft(Gammaleft(frac1nright)^{zeta{left(frac1nright)}}right)$$
which seems converge as shown by Wolfram alpha it's seems converge approximately to $1.6$ But no way to get it's closed form since it's complicated , Now i want the exact value of that sum and it's closed value form if it exists however my narrow idea and attempts are fail to get it .
Note: The Motivation of this question is to know bounds of gamma function power zeta function for $n$ large enough
sequences-and-series gamma-function riemann-zeta
asked Nov 27 '18 at 20:15
zeraoulia rafik
2,38411029
closed as off-topic by user21820, Did, metamorphy, José Carlos Santos, Cesareo Dec 20 '18 at 12:10
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." – Did, metamorphy, José Carlos Santos, Cesareo
If this question can be reworded to fit the rules in the help center, please edit the question.
There at least two errors: $zeta(1/n)$ is not defined for $n=1$ and you put in the wrong formula in Wolfram Alpha (you entered $1/zeta(1/n)$) and therefore the $1.6$ is most probably wrong. Maple gives $1.18$ for $n=2dots 1000$ and $1.205$ for $n=2dots 2000$
– gammatester
Nov 27 '18 at 20:42
Thanks for that , I have edited the starting sum from n=2
– zeraoulia rafik
Nov 27 '18 at 20:44
Replace $zeta(1/n)$ by $zeta(0)+zeta'(0)/n$ and $Gamma(1/n)=n Gamma(1+1/n)$ by $nGamma(1)+Gamma'(1)$ to see if it converges.
– reuns
Nov 27 '18 at 22:19
I think is diverging. If we assume $f(x)=xGamma(x)^{zeta(x)}$, then $lim_{xrightarrow 0}f(x)=lim_{xrightarrow 0}f'(x)=0$, but $lim_{xrightarrow 0}f''(x)=+infty$
– Nikos Bagis
Nov 28 '18 at 0:48
add a comment |
I want to know more about behavior of both Gamma function and zeta function writing them as a power in the form of harmonic series which i got the below form
$$sum_{ngeq 2}dfrac1nleft(Gammaleft(frac1nright)^{zeta{left(frac1nright)}}right)$$
which seems converge as shown by Wolfram alpha it's seems converge approximately to $1.6$ But no way to get it's closed form since it's complicated , Now i want the exact value of that sum and it's closed value form if it exists however my narrow idea and attempts are fail to get it .
Note: The Motivation of this question is to know bounds of gamma function power zeta function for $n$ large enough
sequences-and-series gamma-function riemann-zeta
asked Nov 27 '18 at 20:15
zeraoulia rafik
2,38411029
I want to know more about behavior of both Gamma function and zeta function writing them as a power in the form of harmonic series which i got the below form
$$sum_{ngeq 2}dfrac1nleft(Gammaleft(frac1nright)^{zeta{left(frac1nright)}}right)$$
which seems converge as shown by Wolfram alpha it's seems converge approximately to $1.6$ But no way to get it's closed form since it's complicated , Now i want the exact value of that sum and it's closed value form if it exists however my narrow idea and attempts are fail to get it .
Note: The Motivation of this question is to know bounds of gamma function power zeta function for $n$ large enough
sequences-and-series gamma-function riemann-zeta
sequences-and-series gamma-function riemann-zeta
asked Nov 27 '18 at 20:15
zeraoulia rafik
2,38411029
asked Nov 27 '18 at 20:15
zeraoulia rafik
2,38411029
edited Nov 27 '18 at 21:11
asked Nov 27 '18 at 20:15
zeraoulia rafik
2,38411029
asked Nov 27 '18 at 20:15
zeraoulia rafik
2,38411029
asked Nov 27 '18 at 20:15
zeraoulia rafik
2,38411029
2,38411029
closed as off-topic by user21820, Did, metamorphy, José Carlos Santos, Cesareo Dec 20 '18 at 12:10
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." – Did, metamorphy, José Carlos Santos, Cesareo
If this question can be reworded to fit the rules in the help center, please edit the question.
closed as off-topic by user21820, Did, metamorphy, José Carlos Santos, Cesareo Dec 20 '18 at 12:10
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." – Did, metamorphy, José Carlos Santos, Cesareo
If this question can be reworded to fit the rules in the help center, please edit the question.
There at least two errors: $zeta(1/n)$ is not defined for $n=1$ and you put in the wrong formula in Wolfram Alpha (you entered $1/zeta(1/n)$) and therefore the $1.6$ is most probably wrong. Maple gives $1.18$ for $n=2dots 1000$ and $1.205$ for $n=2dots 2000$
– gammatester
Nov 27 '18 at 20:42
Thanks for that , I have edited the starting sum from n=2
– zeraoulia rafik
Nov 27 '18 at 20:44
Replace $zeta(1/n)$ by $zeta(0)+zeta'(0)/n$ and $Gamma(1/n)=n Gamma(1+1/n)$ by $nGamma(1)+Gamma'(1)$ to see if it converges.
– reuns
Nov 27 '18 at 22:19
I think is diverging. If we assume $f(x)=xGamma(x)^{zeta(x)}$, then $lim_{xrightarrow 0}f(x)=lim_{xrightarrow 0}f'(x)=0$, but $lim_{xrightarrow 0}f''(x)=+infty$
– Nikos Bagis
Nov 28 '18 at 0:48
add a comment |
There at least two errors: $zeta(1/n)$ is not defined for $n=1$ and you put in the wrong formula in Wolfram Alpha (you entered $1/zeta(1/n)$) and therefore the $1.6$ is most probably wrong. Maple gives $1.18$ for $n=2dots 1000$ and $1.205$ for $n=2dots 2000$
– gammatester
Nov 27 '18 at 20:42
Thanks for that , I have edited the starting sum from n=2
– zeraoulia rafik
Nov 27 '18 at 20:44
Replace $zeta(1/n)$ by $zeta(0)+zeta'(0)/n$ and $Gamma(1/n)=n Gamma(1+1/n)$ by $nGamma(1)+Gamma'(1)$ to see if it converges.
– reuns
Nov 27 '18 at 22:19
I think is diverging. If we assume $f(x)=xGamma(x)^{zeta(x)}$, then $lim_{xrightarrow 0}f(x)=lim_{xrightarrow 0}f'(x)=0$, but $lim_{xrightarrow 0}f''(x)=+infty$
– Nikos Bagis
Nov 28 '18 at 0:48
There at least two errors: $zeta(1/n)$ is not defined for $n=1$ and you put in the wrong formula in Wolfram Alpha (you entered $1/zeta(1/n)$) and therefore the $1.6$ is most probably wrong. Maple gives $1.18$ for $n=2dots 1000$ and $1.205$ for $n=2dots 2000$
– gammatester
Nov 27 '18 at 20:42
There at least two errors: $zeta(1/n)$ is not defined for $n=1$ and you put in the wrong formula in Wolfram Alpha (you entered $1/zeta(1/n)$) and therefore the $1.6$ is most probably wrong. Maple gives $1.18$ for $n=2dots 1000$ and $1.205$ for $n=2dots 2000$
– gammatester
Nov 27 '18 at 20:42
Thanks for that , I have edited the starting sum from n=2
– zeraoulia rafik
Nov 27 '18 at 20:44
Thanks for that , I have edited the starting sum from n=2
– zeraoulia rafik
Nov 27 '18 at 20:44
Replace $zeta(1/n)$ by $zeta(0)+zeta'(0)/n$ and $Gamma(1/n)=n Gamma(1+1/n)$ by $nGamma(1)+Gamma'(1)$ to see if it converges.
– reuns
Nov 27 '18 at 22:19
Replace $zeta(1/n)$ by $zeta(0)+zeta'(0)/n$ and $Gamma(1/n)=n Gamma(1+1/n)$ by $nGamma(1)+Gamma'(1)$ to see if it converges.
– reuns
Nov 27 '18 at 22:19
I think is diverging. If we assume $f(x)=xGamma(x)^{zeta(x)}$, then $lim_{xrightarrow 0}f(x)=lim_{xrightarrow 0}f'(x)=0$, but $lim_{xrightarrow 0}f''(x)=+infty$
– Nikos Bagis
Nov 28 '18 at 0:48
I think is diverging. If we assume $f(x)=xGamma(x)^{zeta(x)}$, then $lim_{xrightarrow 0}f(x)=lim_{xrightarrow 0}f'(x)=0$, but $lim_{xrightarrow 0}f''(x)=+infty$
– Nikos Bagis
Nov 28 '18 at 0:48
add a comment |
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There at least two errors: $zeta(1/n)$ is not defined for $n=1$ and you put in the wrong formula in Wolfram Alpha (you entered $1/zeta(1/n)$) and therefore the $1.6$ is most probably wrong. Maple gives $1.18$ for $n=2dots 1000$ and $1.205$ for $n=2dots 2000$
– gammatester
Nov 27 '18 at 20:42
Thanks for that , I have edited the starting sum from n=2
– zeraoulia rafik
Nov 27 '18 at 20:44
Replace $zeta(1/n)$ by $zeta(0)+zeta'(0)/n$ and $Gamma(1/n)=n Gamma(1+1/n)$ by $nGamma(1)+Gamma'(1)$ to see if it converges.
– reuns
Nov 27 '18 at 22:19
I think is diverging. If we assume $f(x)=xGamma(x)^{zeta(x)}$, then $lim_{xrightarrow 0}f(x)=lim_{xrightarrow 0}f'(x)=0$, but $lim_{xrightarrow 0}f''(x)=+infty$
– Nikos Bagis
Nov 28 '18 at 0:48