# dm12-1718.pdf - DM No 12 MPSI3 2017 – 2018 Mardi Exercice...

• No School
• AA 1
• 1

This preview shows page 1 out of 1 page.

MPSI 3 2017 – 2018 DM N o 12 Mardi 27/02/2018 Exercice 1 On considère la suite de polynômes (P n (X)) n N dans R [X] définie par : P 0 (X) = 1 et n N ,P n + 1 (X) = 2XP n (X) - 1 n + 1 (1 + X 2 )P 0 n (X). Q1) a) Calculer P 1 (X) et P 2 (X). b) Montrer que n N ,deg(P n ) 6 n . On note a n le coefficient de X n dans P n (X). c) Montrer que n N , a n + 1 = n + 2 n + 1 a n . En déduire que a n = n + 1. Que dire alors du degré de P n (X)? Q2) Montrer que n N ,P n ( - X) = ( - 1) n P n (X), que dire alors de la parité du polynôme P n (X)? Q3) a) Montrer que n N ,P 0 n + 1 (X) = ( n + 2)P n (X). b) Montrer que n N ,P 2 n + 1 (0) = 0 et P 2 n (0) = ( - 1) n . c) Déduire de ce qui précède que x R ,P n + 1 ( x ) = P n + 1 (0) + ( n + 2) Z x 0 P n ( t ) dt . Calculer ainsi P 3 (X) et P 4 (X). Q4) a) Montrer que n N ,P n + 2 (X) - 2XP n + 1 (X) + (1 + X 2 )P n (X) = 0. b) Soit x R et u n = P n ( x ), on a donc u 0 = 1 et u n + 2 = 2 xu n + 1 - (1 + x 2 ) u n . Exprimer u n en fonction de n et x . c) En déduire : n N ,P n (X) = 1 2 i £ (X + i ) n + 1 - (X - i ) n + 1 / . Q5) a) Montrer que le polynôme P
• Fall '19

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern

Ask Expert Tutors You can ask 0 bonus questions You can ask 0 questions (0 expire soon) You can ask 0 questions (will expire )
Answers in as fast as 15 minutes