If `x gt 0`, `(x^(n))/(1+x+x^(2)+...+x^(2n))` is

The function f (x)={((x ^(2n)))/((x ^(2n) sgn x)^(2n+1))((e ^(1/x)-e ^(-1/x))/(e ^(1/x)+e ^(-(1)/(x))))x ne0 n in N is:

If (1 + x)^(n) = C_(0) + C_(1) x + C_(2) x^(2) + C_(3) x^(3) + … + C_(n) x^(n) , prove that C_(0) - (C_(1))/(2) + (C_(2))/(3) -…+ (-1)^(n) (C_(n))/(n+1) = (1)/(n+1) .

Let f(x) and g(x) be two continuous function and h(x)= lim_(n to oo) (x^(2n).f(x)+x^(2m).g(x))/((x^(2n)+1)). if the limit of h(x) exists at x=1, then one root of f(x)-g(x) =0 is _____.

Statement-1: The middle term of (x+(1)/(x))^(2n) can exceed ((2n)^(n))/(n!) for some value of x. Statement-2: The coefficient of x^(n) in the expansion of (1-2x+3x^(2)-4x^(3)+ . . .)^(-n) is (1*3*5 . . .(2n-1))/(n!)*2^(n) . Statement-3: The coefficient of x^(5) in (1+2x+3x^(2)+ . . .)^(-3//2) is 2.1.

If |(x^n ,x^(n+2), x^(2n)),(1, x^a, a), (x^(n+5), x^(a+6), x^(2n+5))|=0,AAx in R ,w h e r en in N , then value of a is n b. n-1 c. n+1 d. none of these

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

If barx is the mean of x_(1), x_(2), …………., x_(n) , then for a ne 0 , the mean of ax_(1), ax_(2),…….., ax_(n), x_(1)/a, x_(2)/a, ……….., x_(n)/a is

Lt_(x rarr 0) ((1+x)^(n)-nx-1)/(x^(2)) n gt 1 is euqal to

For n gt 0 int_(0)^(2pi)(x sin^(2n)x)/(sin^(2n)x+cos^(2n)x)dx= ….

The equivalent definition of the function f(x)=lim_(n to oo)(x^(n)-x^(-n))/(x^(n)+x^(-n)), x gt 0 , is