let-A-n-m-0-1-x-n-1-x-m-dx-with-n-and-n-integrs-naturals-1-calculate-A-n-m-by-using-factoriels-2-find-n-m-A-nm-

Question Number 53467 by maxmathsup by imad last updated on 22/Jan/19

l e t A_{n m} = \int_{0}^{1} x^{n} {(1 - x)}^{m} d x w i t h n a n d n i n t e g r s n a t u r a l s

1) c a l c u l a t e A_{n m} b y u s i n g f a c t o r i e l s

2) f i n d \sum_{n, m} A_{n m}

Answered by tanmay.chaudhury50@gmail.com last updated on 22/Jan/19

1) b e t a f u n c t i o n \dots

\int_{0}^{1} x^{n + 1 - 1} {(1 - x)}^{m + 1 - 1} d x = \frac{⌈ (n + 1) ⌈ (m + 1)}{⌈ (m + n + 2)}

u s i n g b e t a f u n c t i o n \dots

a n s w e r i s = \frac{n! m!}{(m + n + 1)!}