Find the value of `underset((inejnek))(sum_(i=0)^(oo)sum_(j=0)^(oo)sum_(k=0)^(oo))1/(3^(i)3^(j)3^(k))`.

sum_(i=1)^(oo)sum_(j=1)^(oo)sum_(k=1)^(oo)(1)/(a^(i+j+k)) is equal to (where |a| gt 1 )

Find the sum_(k=1)^(oo) sum_(n=1)^(oo)k/(2^(n+k)) .

Find the value of underset(r = 0) overset(oo)sum tan^(-1) ((1)/(1 + r + r^(2)))

Find sum_(n=1)^(oo)1/(n^2+5n+6)

Find the sum sum_(j=1)^(10)sum_(i=1)^(10)ixx2^(j)

The number of real solution of the equation sin^(-1) (sum_(i=1)^(oo) x^(i +1) -x sum_(i=1)^(oo) ((x)/(2))^(i)) = (pi)/(2) - cos^(-1) (sum_(i=1)^(oo) (-(x)/(2))^(i) - sum_(i=1)^(oo) (-x)^(i)) lying in the interval (-(1)/(2), (1)/(2)) is ______. (Here, the inverse trigonometric function sin^(-1) x and cos^(-1) x assume values in [-(pi)/(2), (pi)/(2)] and [0, pi] respectively)

The value of lim_(n->oo)sum_(r=0)^(n) (sum_(t=0)^(r-1)1/(5^n)*"^n C_r * "^r C_t .(3^t)) is equal to

Find the value of lim_(n rarr oo)sum_(k=1)^(n)(k^(2)+k-1)/((k+1)!) .