n p =0 n + p S = U 0 + U1 + ..... + U n −1 W-./ n د 2 2 ( ∀n ∈ ℕ* ) 2 − ≤ U n ≤ 2 + ن#$%E1 1 1 1 n +1 n T = + + ....... + وP = U 0U1.......U n−1 و U 0 U1 U n −1 1 ( ∀n ∈ ℕ* ) n − n − 1 ≥ ن#'() J#E2 S 2n = U 02 q n −1 ن#$%E1 p =n 1 T ( ∀n ∈ ℕ* ) ∑ ≤ 2 n ن#+ , אJ* n p =1 p S P 2 = ن#+ ,אE2 k =n T n د مSn = ∑U k -./E3
n د مU n =
∑
2
k =1
W5 1 א א
( −1) وU = k =2 n+1 ( −1) W$ 6 אV وU $
6א7 / Vn = ∑ ( n )n ( n )n ∑ n k =2n k =0
k
2k + 1
k
k =0
2k + 1
$ ذ9 (Vn )n ( وU n )n ن#$%E1 k =n
f n ( x ) = ∑
( −1)
k
k = 0 2k + 1
x 2 k +1 -./E2
−1) x 2 n + 2 ( 1 − f ( x ) = ن#$% J# 1 + x2 1 + x2 ( ∀x ∈ ℝ + ) f 2 n +1 ( x ) ≤ arctan x ≤ f 2 n ( x ) ن#5 :# J* n +1
/ n
(Vn )n ( وU n )n $
6< א2/+ , אJج W-%א א א ℝ + x f n ( x ) = x n − n ( x − 1) − 2 -./ 3 =و>و#7<# n د un < 1 < vn ن#50 vn وun $0 (: f n ( x ) = 0 د6نא#$%E1 f n +1 ( x ) − f n ( x ) ?@ABد@س# J#E2 %@( 2/#+ , ( وאun )n
6א%:@د@س# J* −2 −1 ≤ un − 1 ≤ ن#$% Jج n n %@( 2/#+ , ( وאvn )n
6א%:@د@س# J#E3
nlim un د0DE ( ∀n ≥ 3) : →+∞ n
1 1 E ( ∀n ≥ 3) 1 + < 3 G/F ( ∀n ≥ 3) vn > 1 + ن#$% J* n n n ( n − 1) 2 n ( ∀a > 0 ) ( ∀n ∈ ℕ* ) (1 + a ) ≥ 1 + na + a ن#$% J* 2 1 1 ( ∀n ≥ 3) vn < 1 + ن#+ , وאf n 1 + H>0# Jج n n
f x x. k. = +. = â. = +. â. ( ) Ù %$#J# ( ) 1 2 2. 2 2. 1 1. 1 1. n n. n. x. f x. x x. + +. / â. = â. + +. ( ) ( ) ( ) Ù#5:# J* 2 1 2 arctan n n. x f x x f x + â â ⤠⤠R +. ( ) $. 6×> 2 ...
