prove the inequality by induction for the folfowing expression

Question

1+1/2+1/3+1/4+.....+1/2^n>=1+n/2 I gOt stuck after 3rd step because I can't represent 3rd expression with the help of second one as there are other numbers between 1/2^k and 1/2^(K+1)

Answer 1

Suppose that $$S_n=1+{1\over2}+\cdots+{1\over 2^n}>1+{n\over2}.$$

Then:

$$ S_{n+1}=1+{1\over2}+\cdots+{1\over 2^n}+{1\over 2^n+1}+\cdots+{1\over 2^{n+1}}=S_n+{1\over 2^n+1}+{1\over 2^n+2}+\cdots+{1\over 2^{n+1}}. $$ Apart from $S_n$, the rest of the right hand side is the sum of $2^n$ terms, each of them $\ge 1/2^{n+1}$. So we have: $$ S_{n+1}>1+{n\over2}+2^n{1\over 2^{n+1}}=1+{n\over2}+{1\over2}=1+{n+1\over2}. $$