Sum of $50$ terms of $cot^{-1}3+cot^{-1}7+cot^{-1}13+cot^{-1}21+…$ [duplicate]












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  • Explicitly finding the sum of $arctan(1/(n^2+n+1))$

    2 answers





Find the sum of the first $50$ terms of the series $$cot^{-1}3+cot^{-1}7+cot^{-1}13+cot^{-1}21+.....$$




$$
sum_1^{50}=cot^{-1}3+cot^{-1}7+cot^{-1}13+cot^{-1}21+.....\
=tan^{-1}frac{1}{3}+tan^{-1}frac{1}{7}+tan^{-1}frac{1}{13}+tan^{-1}frac{1}{21}+.....=
$$

My reference gives the solution $tan^{-1}dfrac{5}{6}$, but I do not have any clue of doing it ?



Note: I know that $tan^{-1}x+tan^{-1}y=tan^{-1}dfrac{x+y}{1-xy}$ if $xy<1$.










share|cite|improve this question











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marked as duplicate by Lord Shark the Unknown, lab bhattacharjee sequences-and-series
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Dec 16 '18 at 5:15


This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.














  • 1




    $begingroup$
    Note that $$tan^{-1} x+tan^{-1}y=tan^{-1}frac{x+y}{1-xy}$$
    $endgroup$
    – Qurultay
    Dec 15 '18 at 5:33






  • 1




    $begingroup$
    what function is the sequence $3,7,13,21,...$ given by? How are we supposed to find the series if we don't know how to write it in $sum$ notation?
    $endgroup$
    – clathratus
    Dec 15 '18 at 5:36






  • 1




    $begingroup$
    @clathratus, I assume $a_1=3, a_n=a_{n-1}+2n$.
    $endgroup$
    – farruhota
    Dec 15 '18 at 5:41






  • 2




    $begingroup$
    $a_n=1+n+n^2$, I guess
    $endgroup$
    – Claude Leibovici
    Dec 15 '18 at 5:45












  • $begingroup$
    Find the sum of the first $M$ terms, for $M=1,2,3,4,5$ Look for a formula that works for those, and use induction to prove it.
    $endgroup$
    – Empy2
    Dec 15 '18 at 5:51
















1












$begingroup$



This question already has an answer here:




  • Explicitly finding the sum of $arctan(1/(n^2+n+1))$

    2 answers





Find the sum of the first $50$ terms of the series $$cot^{-1}3+cot^{-1}7+cot^{-1}13+cot^{-1}21+.....$$




$$
sum_1^{50}=cot^{-1}3+cot^{-1}7+cot^{-1}13+cot^{-1}21+.....\
=tan^{-1}frac{1}{3}+tan^{-1}frac{1}{7}+tan^{-1}frac{1}{13}+tan^{-1}frac{1}{21}+.....=
$$

My reference gives the solution $tan^{-1}dfrac{5}{6}$, but I do not have any clue of doing it ?



Note: I know that $tan^{-1}x+tan^{-1}y=tan^{-1}dfrac{x+y}{1-xy}$ if $xy<1$.










share|cite|improve this question











$endgroup$



marked as duplicate by Lord Shark the Unknown, lab bhattacharjee sequences-and-series
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Dec 16 '18 at 5:15


This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.














  • 1




    $begingroup$
    Note that $$tan^{-1} x+tan^{-1}y=tan^{-1}frac{x+y}{1-xy}$$
    $endgroup$
    – Qurultay
    Dec 15 '18 at 5:33






  • 1




    $begingroup$
    what function is the sequence $3,7,13,21,...$ given by? How are we supposed to find the series if we don't know how to write it in $sum$ notation?
    $endgroup$
    – clathratus
    Dec 15 '18 at 5:36






  • 1




    $begingroup$
    @clathratus, I assume $a_1=3, a_n=a_{n-1}+2n$.
    $endgroup$
    – farruhota
    Dec 15 '18 at 5:41






  • 2




    $begingroup$
    $a_n=1+n+n^2$, I guess
    $endgroup$
    – Claude Leibovici
    Dec 15 '18 at 5:45












  • $begingroup$
    Find the sum of the first $M$ terms, for $M=1,2,3,4,5$ Look for a formula that works for those, and use induction to prove it.
    $endgroup$
    – Empy2
    Dec 15 '18 at 5:51














1












1








1





$begingroup$



This question already has an answer here:




  • Explicitly finding the sum of $arctan(1/(n^2+n+1))$

    2 answers





Find the sum of the first $50$ terms of the series $$cot^{-1}3+cot^{-1}7+cot^{-1}13+cot^{-1}21+.....$$




$$
sum_1^{50}=cot^{-1}3+cot^{-1}7+cot^{-1}13+cot^{-1}21+.....\
=tan^{-1}frac{1}{3}+tan^{-1}frac{1}{7}+tan^{-1}frac{1}{13}+tan^{-1}frac{1}{21}+.....=
$$

My reference gives the solution $tan^{-1}dfrac{5}{6}$, but I do not have any clue of doing it ?



Note: I know that $tan^{-1}x+tan^{-1}y=tan^{-1}dfrac{x+y}{1-xy}$ if $xy<1$.










share|cite|improve this question











$endgroup$





This question already has an answer here:




  • Explicitly finding the sum of $arctan(1/(n^2+n+1))$

    2 answers





Find the sum of the first $50$ terms of the series $$cot^{-1}3+cot^{-1}7+cot^{-1}13+cot^{-1}21+.....$$




$$
sum_1^{50}=cot^{-1}3+cot^{-1}7+cot^{-1}13+cot^{-1}21+.....\
=tan^{-1}frac{1}{3}+tan^{-1}frac{1}{7}+tan^{-1}frac{1}{13}+tan^{-1}frac{1}{21}+.....=
$$

My reference gives the solution $tan^{-1}dfrac{5}{6}$, but I do not have any clue of doing it ?



Note: I know that $tan^{-1}x+tan^{-1}y=tan^{-1}dfrac{x+y}{1-xy}$ if $xy<1$.





This question already has an answer here:




  • Explicitly finding the sum of $arctan(1/(n^2+n+1))$

    2 answers








sequences-and-series trigonometry inverse-function






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share|cite|improve this question













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edited Dec 15 '18 at 5:45







ss1729

















asked Dec 15 '18 at 5:24









ss1729ss1729

1,9601923




1,9601923




marked as duplicate by Lord Shark the Unknown, lab bhattacharjee sequences-and-series
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Dec 16 '18 at 5:15


This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.









marked as duplicate by Lord Shark the Unknown, lab bhattacharjee sequences-and-series
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Dec 16 '18 at 5:15


This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.










  • 1




    $begingroup$
    Note that $$tan^{-1} x+tan^{-1}y=tan^{-1}frac{x+y}{1-xy}$$
    $endgroup$
    – Qurultay
    Dec 15 '18 at 5:33






  • 1




    $begingroup$
    what function is the sequence $3,7,13,21,...$ given by? How are we supposed to find the series if we don't know how to write it in $sum$ notation?
    $endgroup$
    – clathratus
    Dec 15 '18 at 5:36






  • 1




    $begingroup$
    @clathratus, I assume $a_1=3, a_n=a_{n-1}+2n$.
    $endgroup$
    – farruhota
    Dec 15 '18 at 5:41






  • 2




    $begingroup$
    $a_n=1+n+n^2$, I guess
    $endgroup$
    – Claude Leibovici
    Dec 15 '18 at 5:45












  • $begingroup$
    Find the sum of the first $M$ terms, for $M=1,2,3,4,5$ Look for a formula that works for those, and use induction to prove it.
    $endgroup$
    – Empy2
    Dec 15 '18 at 5:51














  • 1




    $begingroup$
    Note that $$tan^{-1} x+tan^{-1}y=tan^{-1}frac{x+y}{1-xy}$$
    $endgroup$
    – Qurultay
    Dec 15 '18 at 5:33






  • 1




    $begingroup$
    what function is the sequence $3,7,13,21,...$ given by? How are we supposed to find the series if we don't know how to write it in $sum$ notation?
    $endgroup$
    – clathratus
    Dec 15 '18 at 5:36






  • 1




    $begingroup$
    @clathratus, I assume $a_1=3, a_n=a_{n-1}+2n$.
    $endgroup$
    – farruhota
    Dec 15 '18 at 5:41






  • 2




    $begingroup$
    $a_n=1+n+n^2$, I guess
    $endgroup$
    – Claude Leibovici
    Dec 15 '18 at 5:45












  • $begingroup$
    Find the sum of the first $M$ terms, for $M=1,2,3,4,5$ Look for a formula that works for those, and use induction to prove it.
    $endgroup$
    – Empy2
    Dec 15 '18 at 5:51








1




1




$begingroup$
Note that $$tan^{-1} x+tan^{-1}y=tan^{-1}frac{x+y}{1-xy}$$
$endgroup$
– Qurultay
Dec 15 '18 at 5:33




$begingroup$
Note that $$tan^{-1} x+tan^{-1}y=tan^{-1}frac{x+y}{1-xy}$$
$endgroup$
– Qurultay
Dec 15 '18 at 5:33




1




1




$begingroup$
what function is the sequence $3,7,13,21,...$ given by? How are we supposed to find the series if we don't know how to write it in $sum$ notation?
$endgroup$
– clathratus
Dec 15 '18 at 5:36




$begingroup$
what function is the sequence $3,7,13,21,...$ given by? How are we supposed to find the series if we don't know how to write it in $sum$ notation?
$endgroup$
– clathratus
Dec 15 '18 at 5:36




1




1




$begingroup$
@clathratus, I assume $a_1=3, a_n=a_{n-1}+2n$.
$endgroup$
– farruhota
Dec 15 '18 at 5:41




$begingroup$
@clathratus, I assume $a_1=3, a_n=a_{n-1}+2n$.
$endgroup$
– farruhota
Dec 15 '18 at 5:41




2




2




$begingroup$
$a_n=1+n+n^2$, I guess
$endgroup$
– Claude Leibovici
Dec 15 '18 at 5:45






$begingroup$
$a_n=1+n+n^2$, I guess
$endgroup$
– Claude Leibovici
Dec 15 '18 at 5:45














$begingroup$
Find the sum of the first $M$ terms, for $M=1,2,3,4,5$ Look for a formula that works for those, and use induction to prove it.
$endgroup$
– Empy2
Dec 15 '18 at 5:51




$begingroup$
Find the sum of the first $M$ terms, for $M=1,2,3,4,5$ Look for a formula that works for those, and use induction to prove it.
$endgroup$
– Empy2
Dec 15 '18 at 5:51










1 Answer
1






active

oldest

votes


















8












$begingroup$

Hint




  • Use the fact that $cot^{-1}(x)=tan^{-1}frac{1}{x}$


  • $tan^{-1}left(frac{1}{n^2+n+1}right) = tan^{-1}left(frac{(n+1)-n}{1+n(n+1)}right) =tan^{-1}(n+1) -tan^{-1}(n)$







share|cite|improve this answer









$endgroup$




















    1 Answer
    1






    active

    oldest

    votes








    1 Answer
    1






    active

    oldest

    votes









    active

    oldest

    votes






    active

    oldest

    votes









    8












    $begingroup$

    Hint




    • Use the fact that $cot^{-1}(x)=tan^{-1}frac{1}{x}$


    • $tan^{-1}left(frac{1}{n^2+n+1}right) = tan^{-1}left(frac{(n+1)-n}{1+n(n+1)}right) =tan^{-1}(n+1) -tan^{-1}(n)$







    share|cite|improve this answer









    $endgroup$


















      8












      $begingroup$

      Hint




      • Use the fact that $cot^{-1}(x)=tan^{-1}frac{1}{x}$


      • $tan^{-1}left(frac{1}{n^2+n+1}right) = tan^{-1}left(frac{(n+1)-n}{1+n(n+1)}right) =tan^{-1}(n+1) -tan^{-1}(n)$







      share|cite|improve this answer









      $endgroup$
















        8












        8








        8





        $begingroup$

        Hint




        • Use the fact that $cot^{-1}(x)=tan^{-1}frac{1}{x}$


        • $tan^{-1}left(frac{1}{n^2+n+1}right) = tan^{-1}left(frac{(n+1)-n}{1+n(n+1)}right) =tan^{-1}(n+1) -tan^{-1}(n)$







        share|cite|improve this answer









        $endgroup$



        Hint




        • Use the fact that $cot^{-1}(x)=tan^{-1}frac{1}{x}$


        • $tan^{-1}left(frac{1}{n^2+n+1}right) = tan^{-1}left(frac{(n+1)-n}{1+n(n+1)}right) =tan^{-1}(n+1) -tan^{-1}(n)$








        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Dec 15 '18 at 5:51









        crskhrcrskhr

        3,873925




        3,873925















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