1+3+3^(2)+...+3^(x-1)=((3^(n)-1))/(2)

1^(3)+2^(3)+3^(3)+....+n^(3)=((n(n+1))/(2))^(2)

1^(3)+2^(3)+3^(3)+. . .+n^(3)=((n(n+1))/(2))^(2) .

lim_ (n rarr oo) (1+ (1) / (2) + (1) / (2 ^ (2)) + (1) / (2 ^ (3)) + ...... (1) / (2 ^ (n))) / (1+ (1) / (3) + (1) / (3 ^ (2)) + (1) / (3 ^ (3)) ...... (1) / (3 ^ (n)))

(2.3^(n+1)+7.3^(n-1))/(3^(n+1)-2((1)/(3))^(1-n))=

(1)(a+b)+(a^(2)+2b)+(a^(3)+3b)+.... upto n terms =(2)(x+(1)/(x))^(2)+(x^(2)+(1)/(x^(2)))^(2)+(x^(3)+(1)/(x^(3)))^(2)+...(x^(n)+(1)/(x^(n)))^(2)=

1^(3)+2^(3)+3^(3)+.....+n^(3)=(n(n+1)^(2))/(4), n in N

1^(3)+2^(3)+3^(3)+...+n^(3)=n^(2)((n+1)^(2))/(4)

1.3+2.3^(2)+3.3^(3)+............+n.3^(n)=((2n-1)3^(n+1)+3 )/(4)

If L=("lim")_(nvecoo)(2x3^2x2^3x3^4..... x2^(n-1)x3^n)^(1/((n^(2+1))) , then the value of L^4 is

lim_ (x rarr oo) {(1) / (3) + (1) / (3 ^ (2)) + (1) / (3 ^ (3)) + ...... (. 1) / (3 ^ (n))} =