`1+(1)/(3.2^(2))+(1)/(5.2^(4))+(1)/(7.2^(6))+.... `
`(1)log_(e)2 (2)log_(e)3 (3)log_(e)4 (4)log_(e)5 `

(log_(2)3)(log_(1/3)5)(log_(4)6)<1

(log_(5)4)(log_(4)3)(log_(3)2)(log_(2)1) =

If y(x) is solution of differential equation satisfying (dy)/(dx)+((2x+1)/(x))y=e^(-2x),y(1)=(1)/(2)e^(-2) then (A) y(log_(e)2)=log_(e)2(B)y(log_(e)2)=(log_(e)2)/(4)(C)y(x) is decreasing is (0,1)(D)y(x) is decreasing is ((1)/(2),1)

(1)/(log_(2) e) + (1)/(log_(2)e^(2)) + (1)/(log_(2) e^(4)) + .... =

The expression: (((x^(2)+3x+2)/(x+2))+3x-(x(x^(3)+1))/((x+1)(x^(2)+1))-log_(2)8)/((x-1)(log_(2)3)(log_(3)4)(log_(4)5)(log_(5)2)) reduces to

Suppose 1 lt a lt sqrt(e) and log_(e)a^(2)+(log_(e)a^(2))^(2)+(log_(e)a^(2))^(3)+…. =3 [log_(e)a+(log_(e)a)^(2)+(log_(e)a)^(3)+…]" (1)" then log_(e)(a^(3))=

1+((log_(e)n)^(2))/(2!)+((log_(e)n)^(4))/(4!)+...=

If m,n are the roots of the equation x^(2)-x-1=0, then the value of (1+m log_(e)3+((m log_(e)3)^(2))/(2)!+...*)(1+n log_(e)3+((n log_(e)3)^(2))/(2)!+...)/(1+mn log_(e)3+((mn log_(e)3)^(2))/(2)!+......)

Thevalueof int_(1)^((1+sqrt(5))/(2))(x^(2)+1)/(x^(4)-x^(2)+1)log(1+x-(1)/(x))dxis (a) (pi)/(8)(log)_(e)2(b)(pi)/(2)(log)_(e)2(c)-(pi)/(2)(log)_(e)2(d) none of these

The value of 1-log_(e)2+(log_(e)2)^(2)/(2!)-(log_(e)2)^(3)/(3!)+.. is