请问:已知数列{an}的通项公式为an=(2^n)+1,设bn=(2^n)*(an*a(n+1))^-1,求证{bn}的前n项和Sn<1/3
注意:bn后面的a(n+1)中的n+1为角标,表示数列{an}的第n+1项
解：
（1）依题意得：a(n+1)-1=2(an-1),即：[a(n+1)-1]/(an-1)=2
又a1=3,a1-1=2
所以数列{an-1}为首项是2，公比为2的等比数列，有：
an-1=2×2^(n-1)=2^n，
即：an=2^n +1为所求通项公式
（2）证明：
依题意得：bn=2^n/[an×a(n+1)]=2^n/[(2^n +1)×(2^(n+1) +1)]
分子分母同时除以2^n，得：
bn=1/[(1+1/2^n)×(2+1/2^n)]
令t=1/2^n
所以：
bn=1/[(1+1/2^n)×(2+1/2^n)]=1/[(1+t)×(2+t)]= 1/(1+t) - 1/(2+t)
(由于分子在上，分母在下的分式形式无法打出，括号过多又让人看的眼花缭乱，
所以从此处起的1/2全用字母p来表示)
故：
b1= 1/[(1+p^1)×(2+p^1)] =1/( 1+p) - 1/(2+p)
b2= 1/[(1+p^2)×(2+p^2)] < 1/[( 0+p)×(1+p)] =1/( 0+p) - 1/(1+p)
b3= 1/[(1+p^3)×(2+p^3)] < 1/[(-1+p)×(0+p)] =1/(-1+p) - 1/(0+p)
……
bn= 1/[(1+p^n)×(2+p^n)] < 1/[(-(n-2)+p)×(-(n-3)+p)] =1/(-(n-2)+p) - 1/(-(n-3)+p)
将上面n个式子相加，
(会发现其中右边中间的所有项都抵消了,
只剩下第一个式子右边的第二项和最后一个式子右边的第一项)
得：
sn < 1/(-(n-2)+p) - 1/(2+p) (此处开始为了方便，p=1/2)
因为：sn 小于 1/(-(n-2)+p) - 1/(2+p)
所以：sn 小于 1/(-(n-2)+p) - 1/(2+p)的最小值
根据函数的性质可知，当n=1时，1/(-(n-2)+p) - 1/(2+p)有最小值4/15
即：sn < 4/15 < 5/15=1/3
即：sn < 1/3 得证。
（1）依题意得：a(n+1)-1=2(an-1),即：[a(n+1)-1]/(an-1)=2
又a1=3,a1-1=2
所以数列{an-1}为首项是2，公比为2的等比数列，有：
an-1=2×2^(n-1)=2^n，
即：an=2^n +1为所求通项公式
（2）证明：
依题意得：bn=2^n/[an×a(n+1)]=2^n/[(2^n +1)×(2^(n+1) +1)]
分子分母同时除以2^n，得：
bn=1/[(1+1/2^n)×(2+1/2^n)]
令t=1/2^n
所以：
bn=1/[(1+1/2^n)×(2+1/2^n)]=1/[(1+t)×(2+t)]= 1/(1+t) - 1/(2+t)
(由于分子在上，分母在下的分式形式无法打出，括号过多又让人看的眼花缭乱，
所以从此处起的1/2全用字母p来表示)
故：
b1= 1/[(1+p^1)×(2+p^1)] =1/( 1+p) - 1/(2+p)
b2= 1/[(1+p^2)×(2+p^2)] < 1/[( 0+p)×(1+p)] =1/( 0+p) - 1/(1+p)
b3= 1/[(1+p^3)×(2+p^3)] < 1/[(-1+p)×(0+p)] =1/(-1+p) - 1/(0+p)
……
bn= 1/[(1+p^n)×(2+p^n)] < 1/[(-(n-2)+p)×(-(n-3)+p)] =1/(-(n-2)+p) - 1/(-(n-3)+p)
将上面n个式子相加，
(会发现其中右边中间的所有项都抵消了,
只剩下第一个式子右边的第二项和最后一个式子右边的第一项)
得：
sn < 1/(-(n-2)+p) - 1/(2+p) (此处开始为了方便，p=1/2)
因为：sn 小于 1/(-(n-2)+p) - 1/(2+p)
所以：sn 小于 1/(-(n-2)+p) - 1/(2+p)的最小值
根据函数的性质可知，当n=1时，1/(-(n-2)+p) - 1/(2+p)有最小值4/15
即：sn < 4/15 < 5/15=1/3
即：sn < 1/3 得证。
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第1个回答 2011-04-16
four step
; ;
IV remove unwanted files WinXP
Note: If you have hard disk space is really tight, is not recommended if not removed.
1, delete the backup files
click Start → Run → sfc.exe / purgecache
2, remove the XP driver backup
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If you have added new hardware, you must use the XP installation CD.
3, uninstall is not commonly used system components
use Notepad to modify C: windows inf sysoc.inf, use Notepad's Find / Replace function, in search box hide, replace all empty. In this way, put all of the hide is removed, remember to save. Points and then go to Control Panel
4, delete the help files
deleted C: windows Help under the so file.
5, delete the unused input
C: windows ime
If you are using Microsoft's input method, do not delete.
If your WindowsXP optimized through the above, be sure to reset the computer, tidy up the C drive, where I suggest reinstall XP and then be set which is more secure and faster, the effect is more pronounced. Optimization is complete,Discount Oakley sunglasses, you can go to compare before and after optimization of the speed and see if I was wrong to say yes, so what makes you also feel for the old machine!
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3, turn off error reporting
right-click My Computer → Properties → Advanced → error reports, select
4, turn off System Restore
right-click My Computer → Properties → System Restore, Select the
System Restore is eating system resources on the old machine, it must not be
5, closed Automatic Updates
right-click My Computer → Properties → automatic updates, remove the
6, close the Remote Assistance
right-click My Computer → Properties → remote, to off
7, turn off the debugger dr. watson
Click Start → Run → Enter drwtsn32, the addition to the If you have program errors, the hard disk will ring for a long time, but will take a lot of space.
8, turn off the welcome screen when the user logs in the Control Panel, double-click the user
Account → removed
login for this account because the system resources too much.
9, Turn off advanced text services (optional)
in the Control Panel, double-click Regional and Language Options → Languages ??→ Details → Advanced, check And then click Settings → the bottom there is a Switch Removed advanced text services in total memory.
10, shut down WinXP on the ZIP file support
Win
; XP's ZIP support is not as tasteless even for me, because whatever I need not, boot the system to open a zip support, had to idle less and less system resources, a point, click Start → Run, type:
11, turn off Internet time synchronization
double-click the bottom right corner of the time the task bar, click Sync.
;
XP system optimization to make your computer fast
; ;
III turn off unused system services
Note: The following services can be safely shut down,Mac makeup, if you cause the system to shut down the other services can not start,Ray Ban sunglasses, and I can be independent of ah! !
service: Control Panel → Administrative Tools → Services, if you want to turn off a service to double-click it in the
automatic updates windows Microsoft's Automatic Updates service
Error reporting service error reporting
help and support Help and Support Center
IMAPI CD -burning COM service recording services, I think it is a recorder, we will use Nero's
Messenger LAN users, send messages to each other with the
Print Spooler file into memory , re-printing. A printer's friends can not close, ah, will go wrong
Remote Registry Remote Registry operations
Routing and Remote Access on the LAN and WAN environments to provide routing services for enterprises
98 Task Scheduler is the task of the program, it will not
Wireless Zero Configuration wireless card do you have? Did not shut it
System Event Notification System Event tracking service
windows time is over the above mentioned time and date synchronization. Off it. 98 and 2000 do not have this feature, your tables are not allowed to before?
; ;
II System Group Policy and other optimization
1, turn off WinXP's CD-ROM Autorun feature
; in the Start menu Run, type gpedit.msc Computer Configuration → Administrative Templates → System, find the
2, open the reservation of network bandwidth WinXP
Run in the Start menu, type gpedit.msc computer Configuration → Administrative Templates → Network → qos packet scheduler Professional Edition.
3, do not use the IDE port is disabled (Please set)
; Right-click My Computer → Hardware → Device Manager → IDE ATA / ATAPI controllers, double-click the , and set 1. If you only have one device do you disable a device, method is to ; This can greatly speed up the XP boot speed, at least twice as fast, if you do not accidentally drive into a free, and it does not matter to change it back automatically detected to be the
4, DMA mode
open the hard drive right click My Computer → Hardware → Device Manager → IDE ATA / ATAPI controllers, double-click the
You can restart the computer.
5, do not use the device to disable right-click My Computer
→ Hardware → Device Manager, such as: the printer port, COM, LAN (of course you do not use it temporarily under the premise of this device) so that also help conserve system resources.
6, IE cache folder transfer
proposed transfer this folder to another disk, such as D, E in. Debris can reduce the C disk, thus speeding up the system speed.
My Documents can also be transferred to other disk is conducive to backup and security.
; ;
IV remove unwanted files WinXP
Note: If you have hard disk space is really tight, is not recommended if not removed.
1, delete the backup files
click Start → Run → sfc.exe / purgecache
2, remove the XP driver backup
remove driver cache i386 directory driver.cab file.
If you have added new hardware, you must use the XP installation CD.
3, uninstall is not commonly used system components
use Notepad to modify C: windows inf sysoc.inf, use Notepad's Find / Replace function, in search box hide, replace all empty. In this way, put all of the hide is removed, remember to save. Points and then go to Control Panel
4, delete the help files
deleted C: windows Help under the so file.
5, delete the unused input
C: windows ime
If you are using Microsoft's input method, do not delete.
If your WindowsXP optimized through the above, be sure to reset the computer, tidy up the C drive, where I suggest reinstall XP and then be set which is more secure and faster, the effect is more pronounced. Optimization is complete,Discount Oakley sunglasses, you can go to compare before and after optimization of the speed and see if I was wrong to say yes, so what makes you also feel for the old machine!
First, system settings optimized
1, turn off visual effects
vase Right-click My Computer → Properties → Settings → Advanced → performance, first remove all of the hook, retaining only common task > Right-click My Computer → Properties → Advanced → → Startup and Recovery settings, if you only install an operating system, remove all of the hook, if you have multiple operating systems, you can keep the first hook.
3, turn off error reporting
right-click My Computer → Properties → Advanced → error reports, select
4, turn off System Restore
right-click My Computer → Properties → System Restore, Select the
System Restore is eating system resources on the old machine, it must not be
5, closed Automatic Updates
right-click My Computer → Properties → automatic updates, remove the
6, close the Remote Assistance
right-click My Computer → Properties → remote, to off
7, turn off the debugger dr. watson
Click Start → Run → Enter drwtsn32, the addition to the If you have program errors, the hard disk will ring for a long time, but will take a lot of space.
8, turn off the welcome screen when the user logs in the Control Panel, double-click the user
Account → removed
login for this account because the system resources too much.
9, Turn off advanced text services (optional)
in the Control Panel, double-click Regional and Language Options → Languages ??→ Details → Advanced, check And then click Settings → the bottom there is a Switch Removed advanced text services in total memory.
10, shut down WinXP on the ZIP file support
Win
; XP's ZIP support is not as tasteless even for me, because whatever I need not, boot the system to open a zip support, had to idle less and less system resources, a point, click Start → Run, type:
11, turn off Internet time synchronization
double-click the bottom right corner of the time the task bar, click Sync.
;
XP system optimization to make your computer fast
; ;
III turn off unused system services
Note: The following services can be safely shut down,Mac makeup, if you cause the system to shut down the other services can not start,Ray Ban sunglasses, and I can be independent of ah! !
service: Control Panel → Administrative Tools → Services, if you want to turn off a service to double-click it in the
automatic updates windows Microsoft's Automatic Updates service
Error reporting service error reporting
help and support Help and Support Center
IMAPI CD -burning COM service recording services, I think it is a recorder, we will use Nero's
Messenger LAN users, send messages to each other with the
Print Spooler file into memory , re-printing. A printer's friends can not close, ah, will go wrong
Remote Registry Remote Registry operations
Routing and Remote Access on the LAN and WAN environments to provide routing services for enterprises
98 Task Scheduler is the task of the program, it will not
Wireless Zero Configuration wireless card do you have? Did not shut it
System Event Notification System Event tracking service
windows time is over the above mentioned time and date synchronization. Off it. 98 and 2000 do not have this feature, your tables are not allowed to before?
; ;
II System Group Policy and other optimization
1, turn off WinXP's CD-ROM Autorun feature
; in the Start menu Run, type gpedit.msc Computer Configuration → Administrative Templates → System, find the
2, open the reservation of network bandwidth WinXP
Run in the Start menu, type gpedit.msc computer Configuration → Administrative Templates → Network → qos packet scheduler Professional Edition.
3, do not use the IDE port is disabled (Please set)
; Right-click My Computer → Hardware → Device Manager → IDE ATA / ATAPI controllers, double-click the , and set 1. If you only have one device do you disable a device, method is to ; This can greatly speed up the XP boot speed, at least twice as fast, if you do not accidentally drive into a free, and it does not matter to change it back automatically detected to be the
4, DMA mode
open the hard drive right click My Computer → Hardware → Device Manager → IDE ATA / ATAPI controllers, double-click the
You can restart the computer.
5, do not use the device to disable right-click My Computer
→ Hardware → Device Manager, such as: the printer port, COM, LAN (of course you do not use it temporarily under the premise of this device) so that also help conserve system resources.
6, IE cache folder transfer
proposed transfer this folder to another disk, such as D, E in. Debris can reduce the C disk, thus speeding up the system speed.
My Documents can also be transferred to other disk is conducive to backup and security.
第2个回答 2011-04-16
字写得难看希望你能看清看懂 ~~
第3个回答 2011-04-16
an=(2^n)+1 a(n+1)=2^(n+1)+1
bn=(2^n)*(an*a(n+1))^(-1)={[(2^(n+1)+1]-[2^n+1]}/{[(2^n)+1][2^(n+1)+1]}
=1/[2^n+1]-1/[2^(n+1)+1]
b(n-1)=1/[2^(n-1)+1]-1/[2^n+1]
......
b1=1/(2+1)-1/(2^2+1)
上面式子叠加，得
Sn=1/3-1/[2^(n+1)+1]<1/3
得证本回答被提问者采纳
bn=(2^n)*(an*a(n+1))^(-1)={[(2^(n+1)+1]-[2^n+1]}/{[(2^n)+1][2^(n+1)+1]}
=1/[2^n+1]-1/[2^(n+1)+1]
b(n-1)=1/[2^(n-1)+1]-1/[2^n+1]
......
b1=1/(2+1)-1/(2^2+1)
上面式子叠加，得
Sn=1/3-1/[2^(n+1)+1]<1/3
得证本回答被提问者采纳
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