Home
Class 12
MATHS
The value of sum sum(n=1)^(oo)cot^(-1)((...

The value of sum `sum_(n=1)^(oo)cot^(-1)(((n^(2)+2n)(n^(2)+2n+1)+1)/(2n+2))` is equal to If `f(x+y)=f(x)+f(y)-x y-1`, `AA x , `y in R ` and `f(1)=1,` then the number of solution of `f(n)=n` , `n in N ,` is

A

`cos^(-1)((1)/(sqrt(5)))`

B

`sec^(-1)((sqrt(5))/(3))`

C

`sin^(-1)((1)/(sqrt(5)))`

D

`cot^(-1)(1)`

Text Solution

Verified by Experts

The correct Answer is:
C
`T_(n)=tan^(-1)((2n+2)/(1+(n^(2)+2n)(n^(2)+2n+1)))`
`=tan^(-1)((2n+2)/(1+n(n+2)(n+1)(n+1)))`
`=(tan^(-1)(n+1)(n+2)-tan^(-1)n(n+1))`
`therefore S_(n)=sum_(n=1)^(n)T_(n)=(tan^(-1)(n+1)(n+2)-tan^(-1)2)`
So, `lim_(n rarr oo)S_(n)=((pi)/(2)-tan^(-1)2)=cot^(-1)2=sin^(-1)((1)/(sqrt(5)))`
Promotional Banner

Similar Questions

Explore conceptually related problems

If f(x) is a real valued functions satisfying f(x+y) = f(x) +f(y) -yx -1 for all x, y in R such that f(1)=1 then the number of solutions of f(n) = n,n in N , is

If (x+y)=f(x)+f(y)-xy-1AA x,y in R and f(1)=1f(x+y)=f(x)+f(y)-xy-1AA x,y in R and f(1)=1 then the number of solution of f(n)=n,n in N, is 0(b)1 (c) 2 (d) more than 2

If f(x+y)=f(x)+f(y) -xy -1 for all x, y in R and f(1)=1 then f(n)=n, n in N is true if

Let sum_(r=1)^(n) r^(6)=f(n)," then "sum_(n=1)^(n) (2r-1)^(6) is equal to

Let f be a function satisfying f(x+y)=f(x)+f(y) for all x,y in R. If f(1)=k then f(n),n in N is equal to

Let f be a real valued function satisfying f(x+y)=f(x)+f(y) for all x, y in R and f(1)=2 . Then sum_(k=1)^(n)f(k)=

if f(x)=x^(n) then f(1)+f'(1)/(1!)+f''(1)/(2!)+...+(f^(n)(1))/(n!) is equal to

If f_(1)(x)=|x|-2| and f_(2)(x)=|f_(n-1)(x) for all n>=2,n in N, then number of solution of the equation f_(2015)(x)=2 is