求lim(n→∞)(1+1/2+1/4+...+1/2∧n),最后是2的n次方分之一,麻烦写下过程,刚学高数,有点压力~

Question

求lim(n→∞)(1+1/2+1/4+...+1/2∧n),最后是2的n次方分之一,麻烦写下过程,刚学高数,有点压力~

Answer 1

1+1/2+1/4+...+1/2∧n 是首项a1=1,公比q=1/2的等比数列前n+1项的和
∴Sn=a1(1-q^(n+1))/(1-q)
=1(1-(1/2)^(n+1))/(1-1/2)
=2(1-(1/2)^(n+1)
=2-(1/2)^n
∴lim n→∞ [2-(1/2)^n]
=lim n→∞ 2 -lim n→∞(1/2)^n
=2-0
=2