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Find the sum of an infinite geometric se...

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

Text Solution

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`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)`
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