Show that `1/(x+1)+2/(x^2+1)+4/(x^4+1)+…..+2^n/(x^(2n+1))=1 /(x-1)- 2^(n+1)/(x^(2^(n+1)) -1)`

If (1 + x)^(n) = C_(0) + C_(1) x + C_(2) x^(2) + … + C_(n) x^(n) , Show that (2^(2))/(1*2) C_(0) + (2^(3))/(2*3) C_(1) + (2^(4))/(3*4)C_(2) + ...+ (2^(n+2)C_n)/((n+1)(n+2))= (3^(n+2))/((n+1)(n+2))

Show that the middle term in the expansion of (1+x)^(2n) is (1.3.5….(2n-1))/(n!) 2^(n)x^(n) , where n is a positive integer.

Let y_(n)(x) = x^(2) + (x^(2))/(1+x^(2))+(x^(2))/((1+x^(2))^(2))+......(x^(2))/((1+x^(2))^(n-1))and y(x) = underset(n rarr oo)(lim) y_(n) (x) . Discuss the continuity of y_(n)(x)(n = 1, 2, 3....n) and y(x) "at x" = 0

Let y_(n)(x) = x^(2) + (x^(2))/(1+x^(2))+(x^(2))/((1+x^(2))^(2))+......(x^(2))/((1+x^(2))^(n-1))and y(x) = lim_(n rarr oo) y_(n) (x) . Discuss the continuity of y_(n)(x)(n = 1, 2, 3....n) and y(x) "at x" = 0

The coefficient of 1//x in the expansion of (1+x)^n(1+1//x)^n is (a). (n !)/((n-1)!(n+1)!) (b). ((2n)!)/((n-1)!(n+1)!) (c). ((2n)!)/((2n-1)!(2n+1)!) (d). none of these

Find value of (x+(1)/(x))^(3)+(x^(2)+(1)/(x^(2)))^(3)+"........"+(x^(n)+(1)/(x^(n)))^(3) .

Evaluate lim_(xto1)(x+x^(2)+…….+x^(n)-n)/(x-1)

Consider f(x)=(x)/(x^(2)-1)and g(x)=f''(x) Statement I Graph of g(x) is cancave up for xgt1 . Statement II (d^(n))/(dx^(n))f(x)=((-1)^(n)n!)/(2){(1)/((x-1)^(n+1))+(1)/((x+1)^(n+1))}n"inN

If (x + a_(1)) (x + a_(2)) (x + a_(3)) …(x + a_(n)) = x^(n) + S_(1) x^(n-1) + S_(2) x^(n-2) + …+ S_(n) where , S_(1) = sum_(i=0)^(n) a_(i), S_(2) = (sumsum)_(1lei lt j le n) a_(i) a_(j) , S_(3) (sumsumsum)_(1le i ltk le n)a_(i) a_(j) a_(k) and so on . Coefficient of x^(7) in the expansion of (1 + x)^(2) (3 + x)^(3) (5 + x)^(4) is

If y=(1+x)(1+x^2)(1+x^4)(1+x^(2n)), then find (dy)/(dx)a tx=0.