g-x-cos-arctanx-if-g-x-a-n-x-n-determine-the-sequence-a-n-

Question Number 145938 by Mathspace last updated on 09/Jul/21

g (x) = c o s (a r c t a n x)

i f g (x) = Σ a_{n} x^{n} d e t e r m i n e t h e

s e q u e n c e a_{n}

Answered by Olaf_Thorendsen last updated on 09/Jul/21

If x \in R, artan x \in] - \frac{π}{2}, + \frac{π}{2} [

\Rightarrow \cos (\arctan x) > 0

\cos (\arctan x) = + \sqrt{\cos^{2} (\arctan x)}

\cos (\arctan x) = \sqrt{\frac{\cos^{2} (\arctan x)}{\cos^{2} (actan x) + \sin^{2} (\arctan x)}}

\cos (\arctan x) = \frac{1}{\sqrt{1 + \tan^{2} (\arctan x)}} = \frac{1}{\sqrt{1 + x^{2}}}

a_{2 n} = {(- 1)}^{n} \frac{1.3.5 \dots (2 n - 1)}{2.4.6 \dots (2 n)}

a_{2 n} = {(- 1)}^{n} \frac{(2 n)!}{2^{2 n} n!^{2}}

and a_{2 n + 1} = 0