" 125.The value of "lim_(x)|(a_(1)^(1/x)+a_(2)^(1/x)+........+a_(n)^(1/x))/(n))''

lim_(x rarr0)((a_(1)^(x)+a_(2)^(x)......+a_(n)^(x))/(n))^((1)/(x))=

The value of underset(x to 0)"Lt" ((a_(1)^(1//x)+a_(2)^(1//x)+.....+a_(n)^(1//x))/(n))^(nx)) is

Lt_(x to 0)((a_(1)^(x)+a_(2)^(x)+...+a_(n)^(x))/(n))^(1//x)=

lim_(x rarr oo)[((x+a_(1))(x+a_(2))dots.......(x+a_(n)))^((1)/(n))-x]

lim_(x rarr oo){((a_(1))^((1)/(x))+(a_(2))^((1)/(x))+...+(a_(n))^((1)/(x)))/(n)}^(nx)

Lt_(x to oo) ((a_(1)^((1)/(x))+a_(1)^((1)/(x))+....+a_(n)^((1)/(x)))/(n))_(1)^(x)=

(1+x)(sum_(r=0)^(n)nC_(r)x^(r))+(1+x^(2))(sum_(r=0)^(n-1)(n-1)C_(r)x^(r))+...(1+x^(n))(sum_(r=0)^(1)1C_(r)x^(r))=a_(0)+a_(1)x+, then the value of (a_(1))/(a_(n)+1)+(a_(1))/(a_(n))+(a_(2))/(a_(n-1))+......+(a_(n+1))/(a_(0)) is equal to

Given that (1+x+x^(2))^(n)=a_(0)+a_(1)x+a_(2)x^(2)+......+a_(2n)x^(2n) , find the values of a_(0)^(2)-a_(1)^(2)+a_(2)^(2)-a_(3)^(2)+....+a_(2n)^(2)

Evaluate : lim_(x rarr0)((1-cos(a_(1)x)*cos(a_(2)x)*cos(a_(3)x))......cos(a_(n)x))/(x^(2)) where a_(1),a_(2),a_(3).........a_(n)in R

(1+x)^(n)=a_(0)+a_(1)x+a_(2)*x^(2)+......+a_(n)x^(n) then prove that