let f : ]0,1[ contnue integrable u_n =(−1)^n ∫_0 ^1 x^n f(x)dx prove that Σ u


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Question Number 48065 by maxmathsup by imad last updated on 18/Nov/18

$l e t f :] 0, 1 [c o n t n u e i n t e g r a b l e u_{n} = {(- 1)}^{n} \int_{0}^{1} x^{n} f (x) d x$ $p r o v e t h a t Σ u_{n} c n v e r g e a n d f i n d i t s s u m$
Commented by maxmathsup by imad last updated on 23/Nov/18

$w e h a v e \sum_{n = 0}^{\infty} u_{n} = \sum_{n = 0}^{\infty} {(- 1)}^{n} \int_{0}^{1} x^{n} f (x) d x$ $= \int_{0}^{1} (\sum_{n = 0}^{\infty} {(- x)}^{n} f (x)) d x (d u e t o u n i f o r m c o n v e r g e n c e o n [0, 1]$ $= \int_{0}^{1} \frac{f (x)}{1 + x} d x s o Σ u_{n} c o n v e r g e s a n d i t s s u m i s \int_{0}^{1} \frac{f (x)}{1 + x} d x .$
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