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Approximations to series of Ramanujan-type
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Recently I have been playing around with series of the form
$$sum_k=1^inftyfrack^se^kz-1 = sum_k=1^inftysigma_s(k)e^-kz$$
for $s in mathbbZ$ and where $sigma_s(k)$ is the sum of divisors function of order $s$. These series have generated quite a bit of interest over the years, due in large part to some beautiful modular identities of Ramanujan. The most famous example being
$$alpha^-nleft(frac12zeta(2n+1)+sum_k=1^inftyfrack^2n-1e^2alpha k-1right) = \ (-beta)^nleft(frac12zeta(2n+1)+sum_k=1^inftyfrack^2n-1e^2beta k-1right) - 2^2nsum_k=0^n+1(-1)^kfracB_2k(2k)!fracB_2n+2-2k(2n+2-2k)!alpha^n+1-kbeta^k$$
where $alpha,beta > 0, alphabeta=pi^2$ and $B_k$ are the Bernoulli numbers and $zeta(k)$ is the Riemann zeta function. As far as I know there aren't any similar relations or closed forms when $s in 2mathbbZ$. In my investigations I was able to find some approximation formula for general $s > 0$ but which unfortunately perform poorer and poorer as $s rightarrow infty$.
For instance, at $s=2$ we have
$$sum_k=1^inftyfrack^2ze^kz-1 approx frac2zeta(3)z^2 - frac12-fracz24 -sum_j=0^NB^(2)_j+2B_jfracz^j(j+2)!$$
where $B^(k)_n$ are the Norlund polynomials.
I was excited to find this, but then unfortunately realized that since the sum on the right hand side diverges as $N rightarrow infty$ we can only achieve a finite number of accurate digits as the RHS approaches the left from below then surpasses it, growing without bound.
For instance letting $N=37$ we have
$$sum_k=1^inftyfrack^2e^k-1 approx 2 zeta (3)-frac7079280349473240165930796818117208946601102275178567110474102926210628918330759216889856000000000$$
with the right hand side being correct to the 14-th decimal place. This is about the best we can do with the above formula.
I am curious as to whether someone would be able to provide a better approximation. I am not very familiar with sort of thing... so maybe there is a standard way of achieving approximations like the one above?
real-analysis sequences-and-series approximation divergent-series
asked yesterday
FofX
540315
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up vote
3
down vote
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Recently I have been playing around with series of the form
$$sum_k=1^inftyfrack^se^kz-1 = sum_k=1^inftysigma_s(k)e^-kz$$
for $s in mathbbZ$ and where $sigma_s(k)$ is the sum of divisors function of order $s$. These series have generated quite a bit of interest over the years, due in large part to some beautiful modular identities of Ramanujan. The most famous example being
$$alpha^-nleft(frac12zeta(2n+1)+sum_k=1^inftyfrack^2n-1e^2alpha k-1right) = \ (-beta)^nleft(frac12zeta(2n+1)+sum_k=1^inftyfrack^2n-1e^2beta k-1right) - 2^2nsum_k=0^n+1(-1)^kfracB_2k(2k)!fracB_2n+2-2k(2n+2-2k)!alpha^n+1-kbeta^k$$
where $alpha,beta > 0, alphabeta=pi^2$ and $B_k$ are the Bernoulli numbers and $zeta(k)$ is the Riemann zeta function. As far as I know there aren't any similar relations or closed forms when $s in 2mathbbZ$. In my investigations I was able to find some approximation formula for general $s > 0$ but which unfortunately perform poorer and poorer as $s rightarrow infty$.
For instance, at $s=2$ we have
$$sum_k=1^inftyfrack^2ze^kz-1 approx frac2zeta(3)z^2 - frac12-fracz24 -sum_j=0^NB^(2)_j+2B_jfracz^j(j+2)!$$
where $B^(k)_n$ are the Norlund polynomials.
I was excited to find this, but then unfortunately realized that since the sum on the right hand side diverges as $N rightarrow infty$ we can only achieve a finite number of accurate digits as the RHS approaches the left from below then surpasses it, growing without bound.
For instance letting $N=37$ we have
$$sum_k=1^inftyfrack^2e^k-1 approx 2 zeta (3)-frac7079280349473240165930796818117208946601102275178567110474102926210628918330759216889856000000000$$
with the right hand side being correct to the 14-th decimal place. This is about the best we can do with the above formula.
I am curious as to whether someone would be able to provide a better approximation. I am not very familiar with sort of thing... so maybe there is a standard way of achieving approximations like the one above?
real-analysis sequences-and-series approximation divergent-series
asked yesterday
FofX
540315
add a comment |Â
up vote
3
down vote
favorite
up vote
3
down vote
favorite
Recently I have been playing around with series of the form
$$sum_k=1^inftyfrack^se^kz-1 = sum_k=1^inftysigma_s(k)e^-kz$$
for $s in mathbbZ$ and where $sigma_s(k)$ is the sum of divisors function of order $s$. These series have generated quite a bit of interest over the years, due in large part to some beautiful modular identities of Ramanujan. The most famous example being
$$alpha^-nleft(frac12zeta(2n+1)+sum_k=1^inftyfrack^2n-1e^2alpha k-1right) = \ (-beta)^nleft(frac12zeta(2n+1)+sum_k=1^inftyfrack^2n-1e^2beta k-1right) - 2^2nsum_k=0^n+1(-1)^kfracB_2k(2k)!fracB_2n+2-2k(2n+2-2k)!alpha^n+1-kbeta^k$$
where $alpha,beta > 0, alphabeta=pi^2$ and $B_k$ are the Bernoulli numbers and $zeta(k)$ is the Riemann zeta function. As far as I know there aren't any similar relations or closed forms when $s in 2mathbbZ$. In my investigations I was able to find some approximation formula for general $s > 0$ but which unfortunately perform poorer and poorer as $s rightarrow infty$.
For instance, at $s=2$ we have
$$sum_k=1^inftyfrack^2ze^kz-1 approx frac2zeta(3)z^2 - frac12-fracz24 -sum_j=0^NB^(2)_j+2B_jfracz^j(j+2)!$$
where $B^(k)_n$ are the Norlund polynomials.
I was excited to find this, but then unfortunately realized that since the sum on the right hand side diverges as $N rightarrow infty$ we can only achieve a finite number of accurate digits as the RHS approaches the left from below then surpasses it, growing without bound.
For instance letting $N=37$ we have
$$sum_k=1^inftyfrack^2e^k-1 approx 2 zeta (3)-frac7079280349473240165930796818117208946601102275178567110474102926210628918330759216889856000000000$$
with the right hand side being correct to the 14-th decimal place. This is about the best we can do with the above formula.
I am curious as to whether someone would be able to provide a better approximation. I am not very familiar with sort of thing... so maybe there is a standard way of achieving approximations like the one above?
real-analysis sequences-and-series approximation divergent-series
asked yesterday
FofX
540315
Recently I have been playing around with series of the form
$$sum_k=1^inftyfrack^se^kz-1 = sum_k=1^inftysigma_s(k)e^-kz$$
for $s in mathbbZ$ and where $sigma_s(k)$ is the sum of divisors function of order $s$. These series have generated quite a bit of interest over the years, due in large part to some beautiful modular identities of Ramanujan. The most famous example being
$$alpha^-nleft(frac12zeta(2n+1)+sum_k=1^inftyfrack^2n-1e^2alpha k-1right) = \ (-beta)^nleft(frac12zeta(2n+1)+sum_k=1^inftyfrack^2n-1e^2beta k-1right) - 2^2nsum_k=0^n+1(-1)^kfracB_2k(2k)!fracB_2n+2-2k(2n+2-2k)!alpha^n+1-kbeta^k$$
where $alpha,beta > 0, alphabeta=pi^2$ and $B_k$ are the Bernoulli numbers and $zeta(k)$ is the Riemann zeta function. As far as I know there aren't any similar relations or closed forms when $s in 2mathbbZ$. In my investigations I was able to find some approximation formula for general $s > 0$ but which unfortunately perform poorer and poorer as $s rightarrow infty$.
For instance, at $s=2$ we have
$$sum_k=1^inftyfrack^2ze^kz-1 approx frac2zeta(3)z^2 - frac12-fracz24 -sum_j=0^NB^(2)_j+2B_jfracz^j(j+2)!$$
where $B^(k)_n$ are the Norlund polynomials.
I was excited to find this, but then unfortunately realized that since the sum on the right hand side diverges as $N rightarrow infty$ we can only achieve a finite number of accurate digits as the RHS approaches the left from below then surpasses it, growing without bound.
For instance letting $N=37$ we have
$$sum_k=1^inftyfrack^2e^k-1 approx 2 zeta (3)-frac7079280349473240165930796818117208946601102275178567110474102926210628918330759216889856000000000$$
with the right hand side being correct to the 14-th decimal place. This is about the best we can do with the above formula.
I am curious as to whether someone would be able to provide a better approximation. I am not very familiar with sort of thing... so maybe there is a standard way of achieving approximations like the one above?
real-analysis sequences-and-series approximation divergent-series
asked yesterday
FofX
540315
edited 7 hours ago
asked yesterday
FofX
540315
asked yesterday
FofX
540315
asked yesterday
FofX
540315
540315
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