bjr
ex 5
1)
1/(2k - 1) - 1/(2k + 1) =
(réduction au dénominateur commun (2k - 1)(2k + 1)
((2k + 1)/ (2k - 1)(2k + 1)] - [(2k - 1)/ (2k - 1)(2k + 1)] =
[(2k + 1) - (2k - 1)] / (2k - 1)(2k + 1) =
(2k + 1 - 2k + 1) / (2k - 1)(2k + 1) =
2 / (2k - 1)(2k + 1)
2)
soit S la somme à calculer
2S = 2/(5 x 7) + 2/(7 x 9) + ... + 2/(21 X 23)
• 5 = 2 x 3 -1
• 7 = 2 x 3 + 1
d'après le 1) (avec k = 3)
1/5 - 1/7 = 2/(5 x 7)
• 7 = 2 x 4 - 1
• 9 = 2 x 4 + 1
(k = 4)
1/7 - 1/9 = 2/(7 x 9)
.......................
• 21 = 2 x 11 - 1
• 23 = 2 x 11 + 1
(k = 11)
1/21 - 1/23 = 2/(21 x 23)
2S = 1/5 - 1/7 + 1/7 - 1/9 + 1/9 - 1/11 +..... + 1/21 - 1/23
2S = 1/5 - 1/23
=23/(5x23) - 5/5x23)
= 18/(5 x 23)
= 18 / 115
S = 9/115
