1 . 材料一:有理数都能表示成
,(
,且
,s与t互质)的形式,进而有理数集可以表示为{
且
,s与t互质}.
材料二:我们知道.当
时,可以用一次多项式近似表达指数函数,即
;为提高精确度.可以用更高次的多项式逼近指数函数.
设
对等式两边求导,
得![](https://staticzujuan.xkw.com/quesimg/Upload/formula/46fff2ffb69cbf301c9efca778fa2636.png)
对比各项系数,可得:
,
,
,…,
;
所以
,取
,有
,
代回原式:
.
材料三:对于公比为
的等比数列
,当
时,数列
的前n项和
.
阅读上述材料,完成以下两个问题:
(1)证明:无限循环小数3.7为有理数;
(2)用反证法证明:e为无理数(e=2.7182^为自然对数底数).
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/537c64844b32a708d299ff92dc53c747.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0948ca0227d20b76a27cd1a6d65527fb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/823ab696d27d40920c39b8c910789380.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/00957293044aadf33411d25f96a33922.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/823ab696d27d40920c39b8c910789380.png)
材料二:我们知道.当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8ba156ab181b28fa42e7e4596e69c4d3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e8f0237baa1472e643b6654cd8efe601.png)
设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/860870ed643c19574d5d8b3a01b6afca.png)
得
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/46fff2ffb69cbf301c9efca778fa2636.png)
对比各项系数,可得:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a819b1551fee7d49f197b6c7db77a495.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/93dfb46889c0485f74277e329d8c5ec8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/233ede8e2b7ddd6807e67d974b7370ae.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8feaae3fe8a0a3504ce8f2daee1d0a50.png)
所以
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/51190247f6103b03b31a4f6f01420ddf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bb45f673c56a289ea78831c9237e8d20.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8c7717db429760899f23de4d22702543.png)
代回原式:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1b4da092e325d22a89c38348dd5bae89.png)
材料三:对于公比为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6520ff48dba646ba8b7a7d7ae7ca35bc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76aef4cdcb5af742ce28003b7b6c8c20.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a107eb946e0fe41629c644b7628d5cba.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76aef4cdcb5af742ce28003b7b6c8c20.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/12ecdb961754406f92fceddd4f77cfd3.png)
阅读上述材料,完成以下两个问题:
(1)证明:无限循环小数3.7为有理数;
(2)用反证法证明:e为无理数(e=2.7182^为自然对数底数).
您最近一年使用:0次