数列an及正项数列bn满足:a1=0.5,a(n+1)=1除以1+bn,an+bn=1,求bn的通项公式,比较ln(1+
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数列an及正项数列bn满足:a1=0.5,a(n+1)=1除以1+bn,an+bn=1,求bn的通项公式,比较ln(1+bn)与bn的大小
![数列an及正项数列bn满足:a1=0.5,a(n+1)=1除以1+bn,an+bn=1,求bn的通项公式,比较ln(1+](/uploads/image/z/6125950-46-0.jpg?t=%E6%95%B0%E5%88%97an%E5%8F%8A%E6%AD%A3%E9%A1%B9%E6%95%B0%E5%88%97bn%E6%BB%A1%E8%B6%B3%3Aa1%3D0.5%2Ca%EF%BC%88n%2B1%EF%BC%89%3D1%E9%99%A4%E4%BB%A51%2Bbn%2Can%2Bbn%3D1%2C%E6%B1%82bn%E7%9A%84%E9%80%9A%E9%A1%B9%E5%85%AC%E5%BC%8F%2C%E6%AF%94%E8%BE%83ln%EF%BC%881%2B)
a1=1/(1+b1)
b1=1/a1 -1=1/0.5 -1=1
a(n+1)=1/(1+bn)
a(n+1)+b(n+1)=1
a(n+1)=1-b(n+1)
1-b(n+1)=1/(1+bn)
b(n+1)=1- 1/(1+bn)=bn/(1+bn)
1/b(n+1)=(1+bn)/bn=1/bn +1
1/b(n+1)-1/bn=1,为定值.
1/b1=1/1=1
数列{1/bn}是以1为首项,1为公差的等差数列.
1/bn=1+n-1=n
bn=1/n
数列{bn}的通项公式为bn=1/n.
ln(1+bn)=ln(1+ 1/n)=ln[(n+1)/n]=ln(n+1)-ln(n)
[ln(1+bn)]'-bn'
=[ln(n+1)-ln(n)]'-(1/n)'
=[1/(n+1)-1/n]-(-1/n²)
=1/(n+1) -1/n +1/n²
=[n²-n(n+1)+(n+1)]/[n²(n+1)]
=1/[n²(n+1)]>0
即ln(1+bn)比bn减小得慢.
n=1时,ln(1+bn)=ln2 b1=1 b1>ln(1+bn)
n=2时,ln(1+bn)=ln(3/2) bn=1/2 ln(3/2)>1/2
n>2时,ln(1+bn)恒>bn
综上得n=1时,ln(1+bn)bn.
b1=1/a1 -1=1/0.5 -1=1
a(n+1)=1/(1+bn)
a(n+1)+b(n+1)=1
a(n+1)=1-b(n+1)
1-b(n+1)=1/(1+bn)
b(n+1)=1- 1/(1+bn)=bn/(1+bn)
1/b(n+1)=(1+bn)/bn=1/bn +1
1/b(n+1)-1/bn=1,为定值.
1/b1=1/1=1
数列{1/bn}是以1为首项,1为公差的等差数列.
1/bn=1+n-1=n
bn=1/n
数列{bn}的通项公式为bn=1/n.
ln(1+bn)=ln(1+ 1/n)=ln[(n+1)/n]=ln(n+1)-ln(n)
[ln(1+bn)]'-bn'
=[ln(n+1)-ln(n)]'-(1/n)'
=[1/(n+1)-1/n]-(-1/n²)
=1/(n+1) -1/n +1/n²
=[n²-n(n+1)+(n+1)]/[n²(n+1)]
=1/[n²(n+1)]>0
即ln(1+bn)比bn减小得慢.
n=1时,ln(1+bn)=ln2 b1=1 b1>ln(1+bn)
n=2时,ln(1+bn)=ln(3/2) bn=1/2 ln(3/2)>1/2
n>2时,ln(1+bn)恒>bn
综上得n=1时,ln(1+bn)bn.
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