正切函数:
=x+x^3/3+2x^5/15+17x^7/315+62x^9/2835+...+[2^(2n)*(2^(2n)-1)*B(2n-1)*x^(2n-1)]/(2n)!+......(|x|<π/2)【注:B(2n-1)是贝努利数】
扩展资料:
定义:
如果
在点x=x0具有任意阶导数,则幂级数
![](https://video.ask-data.xyz/img.php?b=https://iknow-pic.cdn.bcebos.com/8ad4b31c8701a18b5e9109cf932f07082838fe3c?x-bce-process=image%2Fresize%2Cm_lfit%2Cw_600%2Ch_800%2Climit_1%2Fquality%2Cq_85%2Fformat%2Cf_auto)
称为
在点x0处的泰勒级数。
在泰勒公式中,取x0=0,得到的级数
![](https://video.ask-data.xyz/img.php?b=https://iknow-pic.cdn.bcebos.com/838ba61ea8d3fd1f397bb9ba3d4e251f95ca5f3b?x-bce-process=image%2Fresize%2Cm_lfit%2Cw_600%2Ch_800%2Climit_1%2Fquality%2Cq_85%2Fformat%2Cf_auto)
称为麦克劳林级数。
函数
的麦克劳林级数是x的幂级数,那么这种展开是唯一的,且必然与
的麦克劳林级数一致。
其他常见的麦克劳林级数有
指数函数:
![](https://video.ask-data.xyz/img.php?b=https://iknow-pic.cdn.bcebos.com/5882b2b7d0a20cf4047ee45f7b094b36adaf9947?x-bce-process=image%2Fresize%2Cm_lfit%2Cw_600%2Ch_800%2Climit_1%2Fquality%2Cq_85%2Fformat%2Cf_auto)
自然对数:
![](https://video.ask-data.xyz/img.php?b=https://iknow-pic.cdn.bcebos.com/e1fe9925bc315c606b1c716f80b1cb13485477a0?x-bce-process=image%2Fresize%2Cm_lfit%2Cw_600%2Ch_800%2Climit_1%2Fquality%2Cq_85%2Fformat%2Cf_auto)
几何级数:
![](https://video.ask-data.xyz/img.php?b=https://iknow-pic.cdn.bcebos.com/8644ebf81a4c510fc81df89a6d59252dd52aa550?x-bce-process=image%2Fresize%2Cm_lfit%2Cw_600%2Ch_800%2Climit_1%2Fquality%2Cq_85%2Fformat%2Cf_auto)
正弦函数:
![](https://video.ask-data.xyz/img.php?b=https://iknow-pic.cdn.bcebos.com/a6efce1b9d16fdfafce5dc6eb98f8c5495ee7ba0?x-bce-process=image%2Fresize%2Cm_lfit%2Cw_600%2Ch_800%2Climit_1%2Fquality%2Cq_85%2Fformat%2Cf_auto)
余弦函数:
![](https://video.ask-data.xyz/img.php?b=https://iknow-pic.cdn.bcebos.com/902397dda144ad3468519d07dda20cf430ad8547?x-bce-process=image%2Fresize%2Cm_lfit%2Cw_600%2Ch_800%2Climit_1%2Fquality%2Cq_85%2Fformat%2Cf_auto)
参考资料:百度百科-泰勒级数