已知级数条件收敛,那么级数一般项加绝对值后的级数是发散的,原级数是收敛的。①一般项加绝对值后的级数,先对一般项分子有理化
![](https://video.ask-data.xyz/img.php?b=https://iknow-pic.cdn.bcebos.com/8c1001e93901213fb0721ea144e736d12f2e950d?x-bce-process=image%2Fresize%2Cm_lfit%2Cw_600%2Ch_800%2Climit_1%2Fquality%2Cq_85%2Fformat%2Cf_auto)
然后使用比较审敛法的极限形式,求n趋于无穷大下面的极限
![](https://video.ask-data.xyz/img.php?b=https://iknow-pic.cdn.bcebos.com/d53f8794a4c27d1e8f676bb90bd5ad6eddc4380d?x-bce-process=image%2Fresize%2Cm_lfit%2Cw_600%2Ch_800%2Climit_1%2Fquality%2Cq_85%2Fformat%2Cf_auto)
说明这个级数与级数1/n的(k+1/2)次幂敛散性相同,根据已知条件这是个发散的p级数
![](https://video.ask-data.xyz/img.php?b=https://iknow-pic.cdn.bcebos.com/242dd42a2834349bdc473ef8d9ea15ce36d3be0e?x-bce-process=image%2Fresize%2Cm_lfit%2Cw_600%2Ch_800%2Climit_1%2Fquality%2Cq_85%2Fformat%2Cf_auto)
所以k+1/2≤1,即k≤1/2。②原级数是个交错级数,根据莱布尼茨判别法,要求一般项的绝对值单调递减,分子有理化后可求出是当且仅当k≥-1/2时,随着n增大而减小,同时一般项的绝对值趋于0,当k≥0恒成立,当k<0,一般项绝对值化为
![](https://video.ask-data.xyz/img.php?b=https://iknow-pic.cdn.bcebos.com/1f178a82b9014a90798716c0b9773912b31bee16?x-bce-process=image%2Fresize%2Cm_lfit%2Cw_600%2Ch_800%2Climit_1%2Fquality%2Cq_85%2Fformat%2Cf_auto)
-k<1/2才能保证极限是0,那么k>-1/2。综合①②,得出k的取值范围是
![](https://video.ask-data.xyz/img.php?b=https://iknow-pic.cdn.bcebos.com/2e2eb9389b504fc2c5e7e078f5dde71190ef6d16?x-bce-process=image%2Fresize%2Cm_lfit%2Cw_600%2Ch_800%2Climit_1%2Fquality%2Cq_85%2Fformat%2Cf_auto)