求极限limn→∞(2^(1/2)*2^(1/4)*2^(1/8)*...*2^(1/2n))

题目

求极限limn→∞(2^(1/2)*2^(1/4)*2^(1/8)*...*2^(1/2n))
求极限:limn→∞(2^(1/2)*2^(1/4)*2^(1/8)*...*2^(1/2n))
是求,当n趋近于无穷时,根号2*2开4次方*2开8次方*...*2开2n次方的极限
怎么算得呢?

答案

1/2+1/4+1/8+.1/2^n =1/2*(1-(1/2)^n)/(1-1/2) =1-(1/2)^n limn→∞(2^(1/2)*2^(1/4)*2^(1/8)*...*2^(1/2n)) =lim2^(1/2+1/4+1/8+.+1/2n) =lim2^(1-(1/2)^n) =2^(1-0) =2 我认为最后一项应该是：...*2^(1/2^n)...