实际上,几何级数的概念来源于公比小于1的等比数列,将等比数列前n项求和取极限便是几何级数,其公式为“首项/(1-公比)”,此处分子为1的原因就是首项为1:
1、首先将等比数列的通项公式写出,注意:此处的n从0开始,这也是此题过程中分母为1的主要原因;
![](https://video.ask-data.xyz/img.php?b=https://iknow-pic.cdn.bcebos.com/c75c10385343fbf26673135bbf7eca8065388f6f?x-bce-process=image%2Fresize%2Cm_lfit%2Cw_600%2Ch_800%2Climit_1%2Fquality%2Cq_85%2Fformat%2Cf_auto)
2、用等比数列前n项和公式求等比数列的前n项和,由于n是从0开始,所以等比数列的首项为1;
![](https://video.ask-data.xyz/img.php?b=https://iknow-pic.cdn.bcebos.com/6c224f4a20a44623a5fd101a9722720e0cf3d797?x-bce-process=image%2Fresize%2Cm_lfit%2Cw_600%2Ch_800%2Climit_1%2Fquality%2Cq_85%2Fformat%2Cf_auto)
3、将求出来的等比数列的前n项和进行化简;
![](https://video.ask-data.xyz/img.php?b=https://iknow-pic.cdn.bcebos.com/f603918fa0ec08faf3a859df56ee3d6d55fbda1a?x-bce-process=image%2Fresize%2Cm_lfit%2Cw_600%2Ch_800%2Climit_1%2Fquality%2Cq_85%2Fformat%2Cf_auto)
4、数项级数的前提是公比小于1,这样子才有意义,因此比较ln3和2之间的大小来判断公比是否满足条件;
![](https://video.ask-data.xyz/img.php?b=https://iknow-pic.cdn.bcebos.com/0eb30f2442a7d933930dae0ba24bd11373f00124?x-bce-process=image%2Fresize%2Cm_lfit%2Cw_600%2Ch_800%2Climit_1%2Fquality%2Cq_85%2Fformat%2Cf_auto)
5、对前n项和对n进行取极限,这样便可以得到数项级数的求和公式,便可得到结果。
![](https://video.ask-data.xyz/img.php?b=https://iknow-pic.cdn.bcebos.com/4610b912c8fcc3ceff64834b9d45d688d43f2025?x-bce-process=image%2Fresize%2Cm_lfit%2Cw_600%2Ch_800%2Climit_1%2Fquality%2Cq_85%2Fformat%2Cf_auto)