Find the value of infinite series $sum_{n=1}^{infty} tan^{-1}(2/n^2)$
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Find the value of infinite series
$$ sum_{n=1}^{infty} tan^{-1}left(frac{2}{n^2} right) $$
I tried to find sequence of partial sums and tried to find the limit of that sequence. But I didn't get the answer.
real-analysis sequences-and-series functions limits-without-lhopital
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add a comment |
$begingroup$
Find the value of infinite series
$$ sum_{n=1}^{infty} tan^{-1}left(frac{2}{n^2} right) $$
I tried to find sequence of partial sums and tried to find the limit of that sequence. But I didn't get the answer.
real-analysis sequences-and-series functions limits-without-lhopital
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3
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If I remember correctly, this problem goes back to Ramanujan. And it can be computed using telescoping technique combined with the following identity: $$ tan^{-1}left(frac{1}{n-1}right) - tan^{-1}left(frac{1}{n+1}right) = arctanleft(frac{2}{n^2}right). $$
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– Sangchul Lee
Sep 13 '17 at 7:17
1
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Related : math.stackexchange.com/questions/144944/…, math.stackexchange.com/questions/415512/… math.stackexchange.com/questions/617032/…
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– lab bhattacharjee
Sep 13 '17 at 7:57
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Also an elementary approach is given here. It is works from the vein that @SangchulLee pointed out.
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– Mason
Dec 27 '18 at 15:50
add a comment |
$begingroup$
Find the value of infinite series
$$ sum_{n=1}^{infty} tan^{-1}left(frac{2}{n^2} right) $$
I tried to find sequence of partial sums and tried to find the limit of that sequence. But I didn't get the answer.
real-analysis sequences-and-series functions limits-without-lhopital
$endgroup$
Find the value of infinite series
$$ sum_{n=1}^{infty} tan^{-1}left(frac{2}{n^2} right) $$
I tried to find sequence of partial sums and tried to find the limit of that sequence. But I didn't get the answer.
real-analysis sequences-and-series functions limits-without-lhopital
real-analysis sequences-and-series functions limits-without-lhopital
edited Sep 13 '17 at 7:14
Sangchul Lee
93.6k12167271
93.6k12167271
asked Sep 13 '17 at 7:09
AnanthakrishnanAnanthakrishnan
847
847
3
$begingroup$
If I remember correctly, this problem goes back to Ramanujan. And it can be computed using telescoping technique combined with the following identity: $$ tan^{-1}left(frac{1}{n-1}right) - tan^{-1}left(frac{1}{n+1}right) = arctanleft(frac{2}{n^2}right). $$
$endgroup$
– Sangchul Lee
Sep 13 '17 at 7:17
1
$begingroup$
Related : math.stackexchange.com/questions/144944/…, math.stackexchange.com/questions/415512/… math.stackexchange.com/questions/617032/…
$endgroup$
– lab bhattacharjee
Sep 13 '17 at 7:57
$begingroup$
Also an elementary approach is given here. It is works from the vein that @SangchulLee pointed out.
$endgroup$
– Mason
Dec 27 '18 at 15:50
add a comment |
3
$begingroup$
If I remember correctly, this problem goes back to Ramanujan. And it can be computed using telescoping technique combined with the following identity: $$ tan^{-1}left(frac{1}{n-1}right) - tan^{-1}left(frac{1}{n+1}right) = arctanleft(frac{2}{n^2}right). $$
$endgroup$
– Sangchul Lee
Sep 13 '17 at 7:17
1
$begingroup$
Related : math.stackexchange.com/questions/144944/…, math.stackexchange.com/questions/415512/… math.stackexchange.com/questions/617032/…
$endgroup$
– lab bhattacharjee
Sep 13 '17 at 7:57
$begingroup$
Also an elementary approach is given here. It is works from the vein that @SangchulLee pointed out.
$endgroup$
– Mason
Dec 27 '18 at 15:50
3
3
$begingroup$
If I remember correctly, this problem goes back to Ramanujan. And it can be computed using telescoping technique combined with the following identity: $$ tan^{-1}left(frac{1}{n-1}right) - tan^{-1}left(frac{1}{n+1}right) = arctanleft(frac{2}{n^2}right). $$
$endgroup$
– Sangchul Lee
Sep 13 '17 at 7:17
$begingroup$
If I remember correctly, this problem goes back to Ramanujan. And it can be computed using telescoping technique combined with the following identity: $$ tan^{-1}left(frac{1}{n-1}right) - tan^{-1}left(frac{1}{n+1}right) = arctanleft(frac{2}{n^2}right). $$
$endgroup$
– Sangchul Lee
Sep 13 '17 at 7:17
1
1
$begingroup$
Related : math.stackexchange.com/questions/144944/…, math.stackexchange.com/questions/415512/… math.stackexchange.com/questions/617032/…
$endgroup$
– lab bhattacharjee
Sep 13 '17 at 7:57
$begingroup$
Related : math.stackexchange.com/questions/144944/…, math.stackexchange.com/questions/415512/… math.stackexchange.com/questions/617032/…
$endgroup$
– lab bhattacharjee
Sep 13 '17 at 7:57
$begingroup$
Also an elementary approach is given here. It is works from the vein that @SangchulLee pointed out.
$endgroup$
– Mason
Dec 27 '18 at 15:50
$begingroup$
Also an elementary approach is given here. It is works from the vein that @SangchulLee pointed out.
$endgroup$
– Mason
Dec 27 '18 at 15:50
add a comment |
1 Answer
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It's the argument of the complex number
$$prod_{n=1}^inftyleft(1+frac{2i}{n^2}right)
=prod_{n=1}^inftyfrac{(n-1+i)(n+1-i)}{n^2}
=prod_{n=1}^inftyfrac{(n-1+i)(n^2+2n)}{n^2(n+1+i)}.$$
This infinite product telescopes.
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$begingroup$
Briefly, how did you get the first step? :(
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– Joao Noch
Sep 13 '17 at 7:30
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@JoaoNoch, It follows from $ arctan(x) = arg(1+ix)$. Perhaps the only non-trivial part is that the argument of the product above determines OP's sum only modulo $2pi$. But an estimate $$sum_{n=1}^{infty} arctan(2/n^2) leq frac{pi}{2} + 2(zeta(2)-1) < pi$$ shows that the principal argument function is enough for our purpose.
$endgroup$
– Sangchul Lee
Sep 13 '17 at 7:50
add a comment |
Your Answer
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1 Answer
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1 Answer
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$begingroup$
It's the argument of the complex number
$$prod_{n=1}^inftyleft(1+frac{2i}{n^2}right)
=prod_{n=1}^inftyfrac{(n-1+i)(n+1-i)}{n^2}
=prod_{n=1}^inftyfrac{(n-1+i)(n^2+2n)}{n^2(n+1+i)}.$$
This infinite product telescopes.
$endgroup$
$begingroup$
Briefly, how did you get the first step? :(
$endgroup$
– Joao Noch
Sep 13 '17 at 7:30
$begingroup$
@JoaoNoch, It follows from $ arctan(x) = arg(1+ix)$. Perhaps the only non-trivial part is that the argument of the product above determines OP's sum only modulo $2pi$. But an estimate $$sum_{n=1}^{infty} arctan(2/n^2) leq frac{pi}{2} + 2(zeta(2)-1) < pi$$ shows that the principal argument function is enough for our purpose.
$endgroup$
– Sangchul Lee
Sep 13 '17 at 7:50
add a comment |
$begingroup$
It's the argument of the complex number
$$prod_{n=1}^inftyleft(1+frac{2i}{n^2}right)
=prod_{n=1}^inftyfrac{(n-1+i)(n+1-i)}{n^2}
=prod_{n=1}^inftyfrac{(n-1+i)(n^2+2n)}{n^2(n+1+i)}.$$
This infinite product telescopes.
$endgroup$
$begingroup$
Briefly, how did you get the first step? :(
$endgroup$
– Joao Noch
Sep 13 '17 at 7:30
$begingroup$
@JoaoNoch, It follows from $ arctan(x) = arg(1+ix)$. Perhaps the only non-trivial part is that the argument of the product above determines OP's sum only modulo $2pi$. But an estimate $$sum_{n=1}^{infty} arctan(2/n^2) leq frac{pi}{2} + 2(zeta(2)-1) < pi$$ shows that the principal argument function is enough for our purpose.
$endgroup$
– Sangchul Lee
Sep 13 '17 at 7:50
add a comment |
$begingroup$
It's the argument of the complex number
$$prod_{n=1}^inftyleft(1+frac{2i}{n^2}right)
=prod_{n=1}^inftyfrac{(n-1+i)(n+1-i)}{n^2}
=prod_{n=1}^inftyfrac{(n-1+i)(n^2+2n)}{n^2(n+1+i)}.$$
This infinite product telescopes.
$endgroup$
It's the argument of the complex number
$$prod_{n=1}^inftyleft(1+frac{2i}{n^2}right)
=prod_{n=1}^inftyfrac{(n-1+i)(n+1-i)}{n^2}
=prod_{n=1}^inftyfrac{(n-1+i)(n^2+2n)}{n^2(n+1+i)}.$$
This infinite product telescopes.
answered Sep 13 '17 at 7:21
Lord Shark the UnknownLord Shark the Unknown
104k1160132
104k1160132
$begingroup$
Briefly, how did you get the first step? :(
$endgroup$
– Joao Noch
Sep 13 '17 at 7:30
$begingroup$
@JoaoNoch, It follows from $ arctan(x) = arg(1+ix)$. Perhaps the only non-trivial part is that the argument of the product above determines OP's sum only modulo $2pi$. But an estimate $$sum_{n=1}^{infty} arctan(2/n^2) leq frac{pi}{2} + 2(zeta(2)-1) < pi$$ shows that the principal argument function is enough for our purpose.
$endgroup$
– Sangchul Lee
Sep 13 '17 at 7:50
add a comment |
$begingroup$
Briefly, how did you get the first step? :(
$endgroup$
– Joao Noch
Sep 13 '17 at 7:30
$begingroup$
@JoaoNoch, It follows from $ arctan(x) = arg(1+ix)$. Perhaps the only non-trivial part is that the argument of the product above determines OP's sum only modulo $2pi$. But an estimate $$sum_{n=1}^{infty} arctan(2/n^2) leq frac{pi}{2} + 2(zeta(2)-1) < pi$$ shows that the principal argument function is enough for our purpose.
$endgroup$
– Sangchul Lee
Sep 13 '17 at 7:50
$begingroup$
Briefly, how did you get the first step? :(
$endgroup$
– Joao Noch
Sep 13 '17 at 7:30
$begingroup$
Briefly, how did you get the first step? :(
$endgroup$
– Joao Noch
Sep 13 '17 at 7:30
$begingroup$
@JoaoNoch, It follows from $ arctan(x) = arg(1+ix)$. Perhaps the only non-trivial part is that the argument of the product above determines OP's sum only modulo $2pi$. But an estimate $$sum_{n=1}^{infty} arctan(2/n^2) leq frac{pi}{2} + 2(zeta(2)-1) < pi$$ shows that the principal argument function is enough for our purpose.
$endgroup$
– Sangchul Lee
Sep 13 '17 at 7:50
$begingroup$
@JoaoNoch, It follows from $ arctan(x) = arg(1+ix)$. Perhaps the only non-trivial part is that the argument of the product above determines OP's sum only modulo $2pi$. But an estimate $$sum_{n=1}^{infty} arctan(2/n^2) leq frac{pi}{2} + 2(zeta(2)-1) < pi$$ shows that the principal argument function is enough for our purpose.
$endgroup$
– Sangchul Lee
Sep 13 '17 at 7:50
add a comment |
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$begingroup$
If I remember correctly, this problem goes back to Ramanujan. And it can be computed using telescoping technique combined with the following identity: $$ tan^{-1}left(frac{1}{n-1}right) - tan^{-1}left(frac{1}{n+1}right) = arctanleft(frac{2}{n^2}right). $$
$endgroup$
– Sangchul Lee
Sep 13 '17 at 7:17
1
$begingroup$
Related : math.stackexchange.com/questions/144944/…, math.stackexchange.com/questions/415512/… math.stackexchange.com/questions/617032/…
$endgroup$
– lab bhattacharjee
Sep 13 '17 at 7:57
$begingroup$
Also an elementary approach is given here. It is works from the vein that @SangchulLee pointed out.
$endgroup$
– Mason
Dec 27 '18 at 15:50