analyse

[agm1991]

مساعدة في حل تمرين في التحليل سنة 1
وجزاكم الله خيرا
http://i39.servimg.com/u/f39/14/26/48/74/12311.jpg

[XBEY]

اظن انه هناك خطا في التمرين
http://latex.codecogs.com/gif.latex?%5Cdpi%7B150%7D%200%3C%20a%20%3C%20b

[XBEY]

1
on montre que

http://latex.codecogs.com/gif.latex?%5Cdpi%7B150%7D%20P%28n%29%20%3A%20Un%5C geq%200%20%5C%2C%20%5C%2C%20et%5C%2C%20%5C%2C%20Vn %5Cgeq%200

http://latex.codecogs.com/gif.latex?%5Cdpi%7B150%7D%20P%28o%29%20%3A%20Uo%5C geq%200%20%5C%2C%20%5C%2C%20et%5C%2C%20%5C%2C%20Vo %5Cgeq%200
car a>0 et b>0 supposons P(n) est vraie jusqu'a lordre n

est montrons que P(n+1) est vraie


http://latex.codecogs.com/gif.latex?%5Cdpi%7B120%7D%20U_%7Bn+1%7D%3D%5Csqrt% 7BUnVn%7D%5Cgeq%200%20%5C%2C%20%5C%2C%20car%5C%2C% 20%5C%2C%20Un%5Cgeq%200%5C%2C%20%5C%2C%20%5C%2C%20 et%20%5C%2C%20%5C%2C%20Vn%5Cgeq%200
http://latex.codecogs.com/gif.latex?%5Cdpi%7B120%7D%20V_%7Bn+1%7D%3D%5Cfrac% 7B1%7D%7B2%7D%28Un+Vn%29%5Cgeq%200%20%5C%2C%20%5C% 2C%20car%5C%2C%20%5C%2C%20Un%5Cgeq%200%5C%2C%20%5C %2C%20%5C%2C%20et%20%5C%2C%20%5C%2C%20Vn%5Cgeq%200
donc P(n+1) est vraie
2

on montre

http://latex.codecogs.com/gif.latex?%5Cdpi%7B120%7D%20P%28n%29%20%3A%20Vn%5C geq%20Un

ihttp://latex.codecogs.com/gif.latex?%5Cdpi%7B150%7D%20V_%7Bn+1%7D-%20U_%7Bn+1%7D%3D%5Cfrac%7B1%7D%7B2%7D%28Un+Vn%29-%5Csqrt%7BUnVn%7D%5Cgeq%200
car pour tout x et y reel positive
http://latex.codecogs.com/gif.latex?%5Cdpi%7B150%7D%20%5Cfrac%7Bx+y%7D%7B2%7 D%5Cgeq%20%5Csqrt%7Bxy%7D

http://latex.codecogs.com/gif.latex?%5Cdpi%7B120%7D%20V_%7Bn+1%7D-%20V_%7Bn%7D%3D%5Cfrac%7B1%7D%7B2%7D%28Un-Vn%29-Vn%3D%5Cfrac%7B1%7D%7B2%7D%28Un-Vn%29%5Cleq%200
donc Vn décroissante

http://latex.codecogs.com/gif.latex?%5Cdpi%7B120%7D%20U_%7Bn+1%7D-%20U_%7Bn%7D%3D%5Csqrt%7B%28UnVn%29%7D-Un%5Cgeq%20%5Csqrt%7BUn%5Ctimes%20Un%7D-Un%3D0
donc Un croissante
pour n>1
http://latex.codecogs.com/gif.latex?%5Cleft%20%7C%20Vn-Un%20%5Cright%20%7C%3D%5Cleft%20%7C%20%5Cfrac%7B1% 7D%7B2%7D%28U_%7Bn-1%7D+V_%7Bn-1%7D-2%5Csqrt%7BU_%7Bn-1%7DV_%7Bn-1%7D%7D%29%20%5Cright%20%7C
http://latex.codecogs.com/gif.latex?%5Cleft%20%7C%20Vn-Un%20%5Cright%20%7C%3D%5Cleft%20%7C%20%5Cfrac%7B1% 7D%7B2%7D%28U_%7Bn-1%7D+V_%7Bn-1%7D-2%5Csqrt%7BU_%7Bn-1%7DV_%7Bn-1%7D%7D%29%20%5Cright%20%7C%5Cleq%20%5Cleft%20%7C% 20%5Cfrac%7B1%7D%7B2%7D%28U_%7Bn-1%7D+V_%7Bn-1%7D%29%20%5Cright%20%7C
http://latex.codecogs.com/gif.latex?%5Cleq%20%5Cleft%20%7C%20%5Cfrac%7B1%7D% 7B2%7D%28U_%7Bn-1%7D+V_%7Bn-1%7D%29%20%5Cright%20%7C%5Cleq%20%5Cleft%20%7C%5Cf rac%7B1%7D%7B2%7D%5Cfrac%7B1%7D%7B2%7D%28U_%7Bn-2%7D+V_%7Bn-2%7D%29%5Cright%20%7C%5Cleq%20...%5Cleq%20%5Cleft% 20%7C%28%5Cfrac%7B1%7D%7B2%5E%7Bn%7D%7D%28U_%7B0%7 D+V_%7B0%7D%29%5Cright%7C
http://latex.codecogs.com/gif.latex?0%5Cleq%20%5Cleft%20%7C%20%28Vn-Un%29%20%5Cright%20%7C%5Cleq%20%5Cleft%20%7C%28%5C frac%7B1%7D%7B2%5E%7Bn%7D%7D%28U_%7B0%7D+V_%7B0%7D %29%5Cright%7C




en passage a limite (théoréme gendarme)..
en obtient

http://latex.codecogs.com/gif.latex?%5Cdpi%7B120%7D%20%5Clim_%7Bn%5Crightarr ow%20+%5Cinfty%20%7D%28Vn-Un%29%3D0
donc Vn et Un adjacentes