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)
Asked
Active
Viewed 44 times
0
-
The inequality is false as stated. $1+1/2+1/4<1+\frac{2}{2}$ – Asinomás Sep 15 '15 at 16:27
-
you have a strange series - it starts out as harmonic, but then the last term should be $\frac{1}{n}$ – Alex Sep 15 '15 at 16:27
-
Yes, one of the two sides is wrong, the left side is convergent while the right side is not. But even if we have a harmonic series in the left, it would converge with rate $\log(n)$ while the right is linear. – Asinomás Sep 15 '15 at 16:28
-
This exercise is from Discrete Math Introduction (E. Sheinerman) I dont think that they would post wrong exercise – Arman Piloyan Sep 15 '15 at 16:33
1 Answers
0
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: $$ S_{n+1}>1+{n\over2}+2^n{1\over 2^{n+1}}=1+{n\over2}+{1\over2}=1+{n+1\over2}. $$
Intelligenti pauca
- 50,470
- 4
- 42
- 77