If `N = n!(n in N, n gt 2)`, then find `lim_(N rarr oo)[(log_(2)N)^(-1) + (log_(3)N)^(-1)+…+(log_(n)N)^(-1)]`

lim_(x rarr 0)(x^(n) ln x), n gt 0

If a > 0, c > 0, b = sqrt(ac), ac != 1 and N > 0 , then prove that (log_(a)N)/(log_(c )N) = (log_(a)N - log_(b)N)/(log_(b)N - log_(c )N) .

Evaluate: ("lim")_(n rarr oo)[(n !)/(n^n)]^(1//n)

lim_(nrarroo) [(1)/(n)+(n^(2))/((n+1)^(3))+(n^(2))/((n+2)^(3))+...+(1)/(8n)]

Evaluate : lim_(xrarroo)(1+2+3+...+n)/n^2

Prove that : (vi) log_(a)x + log_(a^2)x^(2) + log_(a^3)x^(3) + ………….+ log_(a^n)x^(n) = log_(a)x^(n)

If log_(b) n = 2 and log_(n) 2b = 2 , then find the value of b.

Evaluate lim_(ntooo) (1)/(n^(2(log_(e)n-log_(e)(n+1)))+n) .

If agt0,cgt0,b=sqrt(ac),a,c and ac ne 1 , N gt0 prove that (log_(a)N)/(log_(c )N)=(log_(a)N-log_(b)N)/(log_(b)N-log_(c )N)

lim_(nrarroo) [(1)/(n+1)+(1)/(n+2)+(1)/(n+3)+...+(1)/(n+n)]