hello-please-give-me-the-limited-development-of-f-in-0-at-3-rd-order-f-x-ln-sin-x-

Question Number 130303 by stelor last updated on 24/Jan/21

hello \dots please give me the limited development of \dots f \dots i n 0 a t 3^{r d} o r d e r .

f (x) = l n (s i n (x))

Commented by stelor last updated on 24/Jan/21

developpement limite en \frac{Π}{2} a l^{'} ordre 3 .

Answered by mathmax by abdo last updated on 24/Jan/21

f (x) = f (\frac{π}{2}) + \frac{x - \frac{π}{2}}{1!} f^{^{'}} (\frac{π}{2}) + \frac{{(x - \frac{π}{2})}^{2}}{2!} f^{(2)} (\frac{π}{2}) + \frac{{(x - \frac{π}{2})}^{3}}{3!} f^{(3)} (0) + {(x - \frac{π}{2})}^{3} ξ (x - \frac{π}{2})

f (\frac{π}{2}) = 0, f^{^{'}} (x) = \frac{cosx}{sinx} \Rightarrow f^{^{'}} (\frac{π}{2}) = 0, f^{(2)} (x) = \frac{- \sin^{2} x - \cos^{2} x}{\sin^{2} x}

= - \frac{1}{\sin^{2} x} \Rightarrow f^{(2)} (\frac{π}{2}) = - 1

f^{(3)} (x) = \frac{2 sinx cosx}{\sin^{4} x} = \frac{2 cosx}{\sin^{3} x} \Rightarrow f^{(3)} (\frac{π}{2}) = 0 \Rightarrow

\ln (sinx) = - \frac{{(x - \frac{π}{2})}^{2}}{2} + o {{(x - \frac{π}{2})}^{3}}

Commented by mathmax by abdo last updated on 24/Jan/21

first line f^{(3)} (\frac{π}{2})

Commented by stelor last updated on 25/Jan/21

merci \dots \dots . . \sqrt{\sqrt{thank}}

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