Absolute convergence of series along lattice points
Let $W$ be a lattice in the complex plane with generators $u, v.$ Define the following series: $E_n = sum_{w in W} (x + w)^{-n}.$ Apparently this series is absolutely convergent for $n geq 3$ but I am unable to see this.
I don't think our choice of $x$ changes absolute convergence so for simplicity I was looking at $sum_{w in W} w^{-3}.$ But for a given $w,$ how many distinct $w in W$ share the same magnitude $|w|?$ I'm guessing this number is bounded by $|w|$ itself so we have that the series is bounded by the normal $p$-series for $p = 2$ which we know converges. Is my reasoning correct?
Furthermore, is it not the same to define $E_n$ as $E_n = lim_{N rightarrow infty} sum_{alpha = -N}^{N}(lim_{M rightarrow infty} sum_{beta = -M}^M (x + alpha u + beta v)^{-n})?$ My book is trying to prove that the series is absolutely convergent for $n = 1, 2$ to show that we can in fact define $E_n$ in such a way. However, it seems odd to me that $sum_{w in W}(x + w)^{-n}$ is actually something else. Should I understand $sum_{w in W}(x + w)^{-n}$ as summing $(x + w)^{-n}$ along $w in W$ with no particular order of the $w$ in mind?
sequences-and-series number-theory
asked 2 days ago
伽罗瓦
1,083615
add a comment |
Let $W$ be a lattice in the complex plane with generators $u, v.$ Define the following series: $E_n = sum_{w in W} (x + w)^{-n}.$ Apparently this series is absolutely convergent for $n geq 3$ but I am unable to see this.
I don't think our choice of $x$ changes absolute convergence so for simplicity I was looking at $sum_{w in W} w^{-3}.$ But for a given $w,$ how many distinct $w in W$ share the same magnitude $|w|?$ I'm guessing this number is bounded by $|w|$ itself so we have that the series is bounded by the normal $p$-series for $p = 2$ which we know converges. Is my reasoning correct?
Furthermore, is it not the same to define $E_n$ as $E_n = lim_{N rightarrow infty} sum_{alpha = -N}^{N}(lim_{M rightarrow infty} sum_{beta = -M}^M (x + alpha u + beta v)^{-n})?$ My book is trying to prove that the series is absolutely convergent for $n = 1, 2$ to show that we can in fact define $E_n$ in such a way. However, it seems odd to me that $sum_{w in W}(x + w)^{-n}$ is actually something else. Should I understand $sum_{w in W}(x + w)^{-n}$ as summing $(x + w)^{-n}$ along $w in W$ with no particular order of the $w$ in mind?
sequences-and-series number-theory
asked 2 days ago
伽罗瓦
1,083615
Dividing the area of the disk by the area of a fundamental parallelogram of $W$ then $# { w in W, |x+w| < r} = O(r^2)$ so $sum_{w in W^*} |x+w|^{-n}$ converges for $n > 2$ in which case the order of summation doesn't matter in $sum_{w in W^*} (x+w)^{-n}$. To understand how the order of summation affects the case $n=1,2$ you can use that $sum_m frac{1}{(x+m)^2} = frac{pi^2}{sin(pi x)^2}$
– reuns
yesterday
add a comment |
Let $W$ be a lattice in the complex plane with generators $u, v.$ Define the following series: $E_n = sum_{w in W} (x + w)^{-n}.$ Apparently this series is absolutely convergent for $n geq 3$ but I am unable to see this.
I don't think our choice of $x$ changes absolute convergence so for simplicity I was looking at $sum_{w in W} w^{-3}.$ But for a given $w,$ how many distinct $w in W$ share the same magnitude $|w|?$ I'm guessing this number is bounded by $|w|$ itself so we have that the series is bounded by the normal $p$-series for $p = 2$ which we know converges. Is my reasoning correct?
Furthermore, is it not the same to define $E_n$ as $E_n = lim_{N rightarrow infty} sum_{alpha = -N}^{N}(lim_{M rightarrow infty} sum_{beta = -M}^M (x + alpha u + beta v)^{-n})?$ My book is trying to prove that the series is absolutely convergent for $n = 1, 2$ to show that we can in fact define $E_n$ in such a way. However, it seems odd to me that $sum_{w in W}(x + w)^{-n}$ is actually something else. Should I understand $sum_{w in W}(x + w)^{-n}$ as summing $(x + w)^{-n}$ along $w in W$ with no particular order of the $w$ in mind?
sequences-and-series number-theory
asked 2 days ago
伽罗瓦
1,083615
Let $W$ be a lattice in the complex plane with generators $u, v.$ Define the following series: $E_n = sum_{w in W} (x + w)^{-n}.$ Apparently this series is absolutely convergent for $n geq 3$ but I am unable to see this.
I don't think our choice of $x$ changes absolute convergence so for simplicity I was looking at $sum_{w in W} w^{-3}.$ But for a given $w,$ how many distinct $w in W$ share the same magnitude $|w|?$ I'm guessing this number is bounded by $|w|$ itself so we have that the series is bounded by the normal $p$-series for $p = 2$ which we know converges. Is my reasoning correct?
Furthermore, is it not the same to define $E_n$ as $E_n = lim_{N rightarrow infty} sum_{alpha = -N}^{N}(lim_{M rightarrow infty} sum_{beta = -M}^M (x + alpha u + beta v)^{-n})?$ My book is trying to prove that the series is absolutely convergent for $n = 1, 2$ to show that we can in fact define $E_n$ in such a way. However, it seems odd to me that $sum_{w in W}(x + w)^{-n}$ is actually something else. Should I understand $sum_{w in W}(x + w)^{-n}$ as summing $(x + w)^{-n}$ along $w in W$ with no particular order of the $w$ in mind?
sequences-and-series number-theory
sequences-and-series number-theory
asked 2 days ago
伽罗瓦
1,083615
asked 2 days ago
伽罗瓦
1,083615
asked 2 days ago
伽罗瓦
1,083615
asked 2 days ago
伽罗瓦
1,083615
asked 2 days ago
伽罗瓦
1,083615
1,083615
Dividing the area of the disk by the area of a fundamental parallelogram of $W$ then $# { w in W, |x+w| < r} = O(r^2)$ so $sum_{w in W^*} |x+w|^{-n}$ converges for $n > 2$ in which case the order of summation doesn't matter in $sum_{w in W^*} (x+w)^{-n}$. To understand how the order of summation affects the case $n=1,2$ you can use that $sum_m frac{1}{(x+m)^2} = frac{pi^2}{sin(pi x)^2}$
– reuns
yesterday
add a comment |
Dividing the area of the disk by the area of a fundamental parallelogram of $W$ then $# { w in W, |x+w| < r} = O(r^2)$ so $sum_{w in W^*} |x+w|^{-n}$ converges for $n > 2$ in which case the order of summation doesn't matter in $sum_{w in W^*} (x+w)^{-n}$. To understand how the order of summation affects the case $n=1,2$ you can use that $sum_m frac{1}{(x+m)^2} = frac{pi^2}{sin(pi x)^2}$
– reuns
yesterday
Dividing the area of the disk by the area of a fundamental parallelogram of $W$ then $# { w in W, |x+w| < r} = O(r^2)$ so $sum_{w in W^*} |x+w|^{-n}$ converges for $n > 2$ in which case the order of summation doesn't matter in $sum_{w in W^*} (x+w)^{-n}$. To understand how the order of summation affects the case $n=1,2$ you can use that $sum_m frac{1}{(x+m)^2} = frac{pi^2}{sin(pi x)^2}$
– reuns
yesterday
Dividing the area of the disk by the area of a fundamental parallelogram of $W$ then $# { w in W, |x+w| < r} = O(r^2)$ so $sum_{w in W^*} |x+w|^{-n}$ converges for $n > 2$ in which case the order of summation doesn't matter in $sum_{w in W^*} (x+w)^{-n}$. To understand how the order of summation affects the case $n=1,2$ you can use that $sum_m frac{1}{(x+m)^2} = frac{pi^2}{sin(pi x)^2}$
– reuns
yesterday
add a comment |
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Dividing the area of the disk by the area of a fundamental parallelogram of $W$ then $# { w in W, |x+w| < r} = O(r^2)$ so $sum_{w in W^*} |x+w|^{-n}$ converges for $n > 2$ in which case the order of summation doesn't matter in $sum_{w in W^*} (x+w)^{-n}$. To understand how the order of summation affects the case $n=1,2$ you can use that $sum_m frac{1}{(x+m)^2} = frac{pi^2}{sin(pi x)^2}$
– reuns
yesterday