补充 三、证明 n4lm(1+-)< 证:设f(x)=lnx,f(x)在n,n+1上满足L定理条件 ∫(n+1)-f(n)=∫(4)(n+1-n) 1(n+1)-m(0=10+3)=r()=1 又n<5<n+1→ n+15 <In(1+-)< n+1
. 1 ) 1 ln(1 1 1 n n n + + 三、证明 证：设f (x) = ln x, f (x)在[n,n + 1]上满足L定理条件。 f (n + 1) − f (n) = f ( )(n + 1− n) 1 ) ( ) 1 ln( + 1) − ln( ) = ln(1+ = f = n n n 又n n + 1 n n 1 1 1 1 + n n n 1 ) 1 ln(1 1 1 + + 补充
四、设f(x)=a1sinx+a2sin2x+a3sin3x+…+ a sinn, 且f(x) <sinx,a1,a2…an为实常数,求证:a1+2a2+…+mmn≤1 证f(x)-f(0)=f(8)x (a, cos 5+2a 2 cos 25+.nan cosn5)x f(x)=a1cos5+2a2cos25+… nan conex≤sinx a, cos 5+2a2 cos 25 +.na, cos n5ssinx-s1 是介于0与x之间,x→0→2→0 imna1cos5+2a2cos25+… na cosns|=a1+2a2+…nn≤1 x→0
( ) sin sin2 sin3 sin , 四、设f x = a1 x + a2 x + a3 x ++ an nx 且 f (x) sin x ,a1 ,a2 an为实常数， 2 1. 求证：a1 + a2 ++ nan 证 f (x)− f (0) = f ()x = (a1 cos + 2a2 cos2 +nan cosn )x f (x) = a1 cos + 2a2 cos2 +nan cosn x sin x 1 1 1 cos + 2 2 cos2 + cos sin x a a nan n x 是介于0与x之间，x → 0 → 0. lim 1 cos 2 2 cos2 cos 1 2 2 1 0 + + = + + → n n x a a na n a a na