let-f-0-1-contnue-integrable-u-n-1-n-0-1-x-n-f-x-dx-prove-that-u-n-cnverge-and-find-its-sum-

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 .