题目内容
已知函数f(x)满足f(x+y)=f(x)·f(y),且f(1)=
.
(1)当n∈N*时,求f(n)的表达式;
(2)当an=n·f(n), n∈N*,求证a1+a2+…+an<2;
(3)设bn=
.
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823161244967225.gif)
(1)当n∈N*时,求f(n)的表达式;
(2)当an=n·f(n), n∈N*,求证a1+a2+…+an<2;
(3)设bn=
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408231612449831928.gif)
(1)f(n)}为首页f(n)=![](http://thumb.1010pic.com/pic2/upload/papers/20140823/2014082316124499873.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408231612450291132.gif)
(2)略
(3)
=![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823161245061308.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/2014082316124499873.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408231612450291132.gif)
(2)略
(3)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823161245045590.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823161245061308.gif)
(1)解:令x=n,y=1得,f(n+1)=f(n)·f(1),
∴{f(n)}为首页f(n)=![](http://thumb.1010pic.com/pic2/upload/papers/20140823/2014082316124499873.gif)
.
(2)证明an=n·f(n),= n·(
)n,由错位相减法证得a1+a2+a3+…+an=2-
;
(3)解:∵
,
∴
.
故
.
∴![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823161245045590.gif)
=![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408231612452631013.gif)
=![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823161245061308.gif)
∴{f(n)}为首页f(n)=
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/2014082316124499873.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408231612450291132.gif)
(2)证明an=n·f(n),= n·(
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823161244967225.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823161245154790.gif)
(3)解:∵
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823161245185550.gif)
∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823161245201839.gif)
故
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408231612452321071.gif)
∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823161245045590.gif)
=
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408231612452631013.gif)
=
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823161245061308.gif)
![](http://thumb2018.1010pic.com/images/loading.gif)
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