Question

求大神解数学题





Answer 1

1/(n^2+n+1)+2/(n^2+n+2)+...+n/(n^2+n+n)
≤(1+2+...+n)/(n^2+n+1)
=n(n+1)/[2(n^2+n+1)] (1)
----------
1/(n^2+n+1)+2/(n^2+n+2)+...+n/(n^2+n+n)
≤(1+2+...+n)/(n^2+n+n)
=n(n+1)/[2(n^2+n+n)] (2)
-----------------
lim(n->∞) n(n+1)/[2(n^2+n+1)] =1/2
lim(n->∞) n(n+1)/[2(n^2+n+n)] =1/2
=>lim(n->∞) [1/(n^2+n+1)+2/(n^2+n+2)+...+n/(n^2+n+n)] =1/2
(2)
lim(x->0) { √[ 1+ (1/x)f(x)] -1 }/ x^2 = c
x->0
f(x) ~ τx^k
-----------
solution:
x->0
1+ (1/x)f(x) ~ 1+ τx^(k-1)
√[ 1+ (1/x)f(x) ] ~ 1 + (1/2)τx^(k-1)
√[ 1+ (1/x)f(x) ] -1 ~ (1/2)τx^(k-1)
lim(x->0) { √[ 1+ (1/x)f(x)] -1 }/ x^2 = c
=>
τ-1 = 2
τ=3
and
(1/2)τ = c
τ= 2c

Answer 2

如图

追答