limn→∞1+12+122+…+12n1−12+122−…+(−1)n•12n=．

Question

lim

n→∞

(1+(1)/(2)+(1)/(22)+…+(1)/(2n))/(1−(1)/(2)+(1)/(22)−…+(−1)n•(1)/(2n))=______．

Answer 1

当n为偶数时，

lim

n→∞

(1+(1)/(2)+(1)/(22)+…+(1)/(2n))/(1−(1)/(2)+(1)/(22)−…+(−1)n•(1)/(2n))
=

lim

n→∞

((1×(1−((1)/(2))n))/(1−(1)/(2)))/((1+(1)/(22)+(1)/(24)+…+(1)/(2n)) −((1)/(2)+(1)/(23)+…+(1)/(2n−1)))
=

lim

n→∞

(2−(2)/(2n))/((1−(1)/(2(n)/(2)))/(1−(1)/(2))−((1)/(2)(1−(1)/(2(n)/(2))))/(1−(1)/(2)))
=

lim

n→∞

(2−(2)/(2n))/(作业帮用户2017-10-07问题解析根据题意，分2种情况，当n为偶数时，原式=limn→∞((1×(1−((1)/(2))n))/(1−(1)/(2)))/((1+(1)/(22)+(1)/(24)+…+(1)/(2n)) −((1)/(2)+(1)/(23)+…+(1)/(2n−1)))；当n为奇数时，原式=limn→∞((1−(1)/(2n))/(1−(1)/(2)))/((1+(1)/(22)+(1)/(24)+…+(1)/(2n−1)) −((1)/(2)+(1)/(23)+…+(1)/(2n)))．由此可求出limn→∞(1+(1)/(2)+(1)/(22)+…+(1)/(2n))/(1−(1)/(2)+(1)/(22)−…+(−1)n•(1)/(2n))的值．名师点评本题考点：极限及其运算；数列的求和．考点点评：本题考查数列的极限，解题时要注意培养计算能力．扫描下载二维码©2018 作业帮 联系方式：[e-mail]作业帮协议)