电路中的f=0x1=0什么意思

2024-11-07 09:31:56
推荐回答(1个)
回答(1):

(1)令x1=x2=0,得f(0)=f(x0)+2f(0),∴f(x0)=-f(0).①
令x1=1,x2=0,得f(x0)=f(x0)+f(1)+f(0),∴f(1)=-f(0).②
由①②得 f(x0)=f(1).∴f(x)为单调函数,
∴x0=1.
(2)由(1)得f(x1+x2)=f(x1)+f(x2)+f(1)=f(x1)+f(x2)+1.
∵f(n+1)=f(n)+f(1)+1=f(n)+2,f(1)=1,∴f(n)=2n-1.(n∈Z*)
∴an=
1
2n?1

又∵f(1)=f(
1
2
+
1
2
)=f(
1
2
)+f(
1
2
)+f(1)
∴f(
1
2
)=0,b1=f(
1
2
)+1.
又f(
1
2n
)=f(
1
2n+1
+
1
2n+1
)=f(
1
2n+1
)+f(
1
2n+1
)+f(1)=2f(
1
2n+1
)+1,
∴2bn+1=2f(
1
2n+1
)+2=f(
1
2n
)+1=bn.
∴bn=(
1
2
)n?1.
∴Sn=
1
1×3
+
1
3×5
+…+
1
(2n?1)(2n+1)

1
2
(
1
1
?
1
3
+
1
3
?
1
5
+…+
1
2n?1
?
1
2n+1
)
=
1
2
(1?
1
2n+1
)
Tn=(
1
2
)0(
1
2
)1+(
1
2
)1(
1
2
)2+…+(
1
2
)n?1(
1
2
)n=
1
2
+(
1
2
)3+…+(
1
2
)2n?1
=

1
2
[1?(
1
4
)n]

1?
1
4


2
3
[1?(
1
4
)n].

4
3
Sn?Tn=
2
3
(1?
1
2n+1
)?
2
3
[1?(
1
4
)n]=
2
3
[(
1
4
)n?
1
2n+1
].
∵4n=(3+1)n=Cnn3n+Cnn-13n-1+…+Cn13+Cn0≥3n+1>2n+1,

4
3
Sn?Tn=
3
2
(
1
4n
?
1
2n+1
)<0.

4
3
Sn<Tn.