Find the sum `1+(1)/(1+2) + (1)/(1+2+3) + cdots+ (1)/(1+2+3+cdots+ n )`

Find the sum 1^(2) + (1^(2)+2^(2))+ (1^(2)+2^(2)+3^(2))+ cdots up to 22 nd term.

Find the sum of the series 1.n + 2 .(n-1) + 3.(n-2) + cdots + ( n-1) .2 + n.1.

If H_(n) =1+(1)/(2)+ cdots + (1)/(n) then value of S_(n) =1 + (3)/(2) + (5)/(3) +cdots + (2n-1)/(n) is

If S denotes the sum to infinity and S_(n) , denotes the sum of n terms of the series 1+(1)/(2)+(1)/(4) + (1)/(8)+ cdots , such that S-S_(n) lt (1)/(100) , then the least value of n is

Find the sum of the infinite series (1)/(9) + (1)/(18) +(1)/(30) +(1)/(45)+(1)/(63)+cdots.

Let x , |x-1| and |x + 1| be the first three terms of a G.P. then sum of the series (1)/(x)+ (1)/(|x-1|)+(1)/(|x+1|)+cdots oo is equal to

The sum of the series tan ^(-1)"" (1)/( 1 + 1 + 1 ^(2)) + tan ^(-1)"" (1)/( 1 + 2 + 2 ^(2)) + tan ^(-1) ""(1)/( 1 + 3 + 3 ^(2)) +... is equal to : a) pi/4 b) pi/2 c) pi/3 d) pi/6

Find the sum to n terms of the series 1//(1xx2) + 1//(2xx3) + 1//(3xx4) + cdots + 1//n(n+1).

If the sum to infinity of the series 3+(3+d) (1)/(4) +(3+2d) (1)/(4^(2))+cdots oo is (44)/(9) then find d.

Let a_(1), a_(2),a_(3), cdots , a_(4001) are in A.P. such that (1)/(a_(1)a_(2))+(1)/(a_(2)a_(3))+ cdots + (1)/(a_(4000) a_(4001))= 10 and a_(2)+ a_(4000) = 50 then find the value of |a_(1)-a_(4001)| .