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[" 44.For any positive integer "n," defi...

[" 44.For any positive integer "n," define "f_(n):(0,oo)rarr R" as "],[f_(n)(x)=sum_(j=1)^(n)tan^(-1)((1)/(1+(x+j)(x+j-1)))" for all "x in(0,oo)],[" (Here,the inverse trigonometric function "tan^(-1)x],[" (Here,the inverse trigonometric function "tan^(-1)x],[" assumes values in "(-(pi)/(2),(pi)/(2))" ).Then,which of the "],[" following statement(s) is (are) TRUE? "],[[" (a) "sum_(j=1)^(5)tan^(2)(f_(j)(0))sec^(2)(f_(j)(0))=10,],[" (c) For any fixed positive integer "n,lim_(x rarr oo)sec^(2)(f_(n)(x))=1,]]

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For any positive integer n , define f_n :(0,oo)rarrR as f_n(x)=sum_(j=1)^ntan^(-1)(1/(1+(x+j)(x+j-1))) for all x in (0, oo) . Here, the inverse trigonometric function tan^(-1)x assumes values in (-pi/2,pi/2)dot Then, which of the following statement(s) is (are) TRUE? sum_(j=1)^5tan^2(f_j(0))=55 (b) sum_(j=1)^(10)(1+fj '(0))sec^2(f_j(0))=10 (c) For any fixed positive integer n , (lim)_(xrarroo)tan(f_n(x))=1/n (d) For any fixed positive integer n , (lim)_(xrarroo)sec^2(f_n(x))=1

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