For every natural number 1/1 + 1/2 + 1/3 + ... + 1/((2^n)-1)+1/2^n \(\ge\) 1+ n/2

 For every natural number n,

 1/1 + 1/2 + 1/3 + ... + 1/((2^n)-1)+1/2^n \(\ge\) 1+ n/2

Base Case: n=1

1/(2^1-1)+1/2 \(\ge\) 1 + 1/2

3/2 \(\ge\) 3/2

Base case holds.

Induction Step:

Assume induction hypothesis in base case holds for n, then prove it holds for n+1 as well.

1/1 + 1/2 + 1/3 + ... + 1/((2^n+1)-1)+1/2^(n+1) \(\ge\) 1+ (n+1)/2

1 +  n/2 + 1/(2^(n+1)-1) + 1/2^(n+1) \(\ge\) 1 + (n+1)/2

1 + (n(2^(n+1)-1)/(2(2^(n+1)-1)) + 2/2(2^(n+1)-1) + 1/2^(n+1) \(\ge\) 1 + (n+1)/2

1 + (n((2^(n+1)-1)+2)/(2(2^(n+1)-1)) + 1/2^(n+1) \(\ge\) 1 + (n+1)/2

1+ (n+2)/2 + 1/2(n+1) \(\ge\) 1 + (n+1)/2

which is true, and proves our induction hypothesis.