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find-the-sum-of-the-infinite-series-tan-1-2-n-2-




Question Number 30235 by NECx last updated on 18/Feb/18
find the sum of the infinite  series        tan^(−1) ((2/n^2 ))
findthesumoftheinfiniteseriestan1(2n2)
Commented by prof Abdo imad last updated on 18/Feb/18
for n≥1   (2/n^2 )= ((n+1−(n−1))/(1+(n+1)(n−1))) let put n=tanu_n   (2/n^2 )= ((tanu_(n+1) −tan_(n−1) )/(1+tanu_n tanu_(n−1) ))=tan(u_(n+1) −u_(n−1) )  arctan((2/n^2 ))= u_(n+1)  −u_(n−1 )  and  S_N =Σ_(n=1) ^N arctan((2/n^2 ))=Σ_(n=1) ^N ((u_(n+1) −u_n ) +(u_n −u_(n−1) ))  =Σ_(n=1) ^N (u_(n+1)  −u_n ) +Σ_(n=1) ^N (u_n  −u_(n−1) )  =u_(N+1)  −u_1   +u_N  −u_(0 ) =arctan(N+1)+arctanN −(π/4)  lim_(N→∞)  S_N = (π/2) +(π/2) −(π/4)= ((3π)/4) .
forn12n2=n+1(n1)1+(n+1)(n1)letputn=tanun2n2=tanun+1tann11+tanuntanun1=tan(un+1un1)arctan(2n2)=un+1un1andSN=n=1Narctan(2n2)=n=1N((un+1un)+(unun1))=n=1N(un+1un)+n=1N(unun1)=uN+1u1+uNu0=arctan(N+1)+arctanNπ4limNSN=π2+π2π4=3π4.
Commented by abdo imad last updated on 18/Feb/18
arctan means tan^(−1) .
arctanmeanstan1.
Commented by NECx last updated on 19/Feb/18
thank you so much
thankyousomuch

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