0

Is it possible to prove $$1 + \frac{k}{2} + \frac{1}{2^{k+1}} \ge 1+ \frac{k+1}{2}\ \text{?}$$

Michael Hardy
  • 1
  • This is false when $k>1$. – T.J. Gaffney Dec 18 '15 at 21:39
  • What number proves the following equation wrong? $$1+\frac{1}{2} + \frac{1}{3} + \dots + \frac{1}{2^n -1} + \frac{1}{2^n} \ge 1 + \frac{n}{2}$$ where n is a positive integer –  Dec 19 '15 at 00:50

1 Answers1

1

This is false. Take $k=1$. The LHS becomes $1.75$ and the RHS becomes $2$. It can easily be proven that the negation holds: $1+\frac{k}{2}+\frac{1}{2^{k+1}}<1+\frac{k+1}{2}$. Just break up the fraction on the right and you see $$1+\frac{k}{2}+\frac{1}{2^{k+1}}<1+\frac{k}{2} +\frac{1}{2}$$ $$\frac{1}{2^{k+1}}<\frac{1}{2}$$ which is obviously true for $k>0$. (Of course this is not a formal proof. To proof it, one would work backwards from the last statement.)

Trevor Norton
  • 1,603