Find the sum of an infinite geometric series whose first term is the limit of the function `f(x)=(tan x-sin x)/(sin^3x)` as `x->0` and whose common ratio is the limit of the function `g(x) =(1-sqrt(x))/(cos^(-1)x)^2` as x->1

Find the sum of an infinite geometric series whose first term is the l

1.	Find the sum of an infinite geometric series whose first term is the limit of the function `f(x)=(tan x-sin x)/(sin^3x)` as `x->0` and whose common ratio is the limit of the function `g(x) =(1-sqrt(x))/(cos^(-1)x)^2` as x->1
Answer» `a=lim_(x->0)(tanx-sinx)/(sin^3x)` `=lim_(x->0)(sinx(secx-1))/(sin^3x)` `=lim_(x->0)(seccx-1)/sin^2x(secx+1)/(secx+1)` `=lim_(x->0)(sec^2x-1)/(sin^2x(secx-1))` `=lim_(x->0)(tan^2x)/(sin^2x(secx+1))` `=(sec^2 0)/(sec 0+1)=1/(1+1)=1/2` `a=1/2` `r=lim_(x->1)(1-sqrtx)/(cos^(-1)x)^2(1+sqrtx)/(1+sqrtx)` `1/(1+sqrt1)lim_(x->1)(1-x)/(cos^(-1)x)^2` `1/2lim_(t->0)(1-cost)/t^2` `1/2lim_(t->0)(2sin^2(t/2))/t^2` `lim_(t->0)1/4((sin(t/2))/(t/2))((sin(t/2))/(t/2))` `=1/4``a=1/2,r=1/4` `s_oo=a/(1-r)=(1/2)/(1-1/4)=1/2*4/3=2/3.`