we-define-the-bernoulli-polynomial-B-n-by-b-0-1-and-n-N-b-n-n-b-n-1-and-0-1-b-n-t-dt-0-1-find-b-n-1-b-n-0-for-n-2-2-prove-that-b-n-x-1-n-b-n-1-x-n-N-3-calculate-b-0

Question Number 30483 by abdo imad last updated on 22/Feb/18

w e d e f i n e t h e b e r n o u l l i p o l y n o m i a l B_{n} b y

b_{0} = 1 a n d \forall n \in N^{★} b_{n}^{^{'}} = n b_{n - 1} a n d \int_{0}^{1} b_{n} (t) d t = 0

1) f i n d b_{n} (1) - b_{n} (0) f o r n ⩾ 2

2) p r o v e t h a t b_{n} (x) = {(- 1)}^{n} b_{n} (1 - x) \forall n \in N

3) c a l c u l a t e b_{0}, b_{1}, b_{2}, b_{3}