Discrete Mathematics Proving Question

Question

I need help with this proof. I am stuck on this question and don't know how to do it:

Prove that:

$$ \forall n \in \mathbb{N}, \sum\limits_{i=2}^{2^n} \frac{1}{i} \geq \frac{n}{2}$$

Possible duplicate of Why does the series $\sum_{n=1}^\infty\frac1n$ not converge? — colormegone, Dec 04 '15 at 07:22

score 3 · Answer 1 · answered Dec 04 '15 at 01:00

3

Hint: Show that the following is true:

$$\underbrace{\frac12}_{\geq \frac12} + \underbrace{\frac13+\frac14}_{\geq \frac12} + \underbrace{\frac15+\frac16+\frac17+\frac18}_{\geq \frac12}+\ldots + \underbrace{\frac1{2^{n-1}+1}+\ldots+\frac1{2^n}}_{\geq \frac12}$$

answered Dec 04 '15 at 01:00

Théophile

24,627

1

Great explanation, great visual! – orangeskid Dec 04 '15 at 01:06
I may sound stupid, but what significance does that have? – Johnny Dec 04 '15 at 01:19
@Johnny — There are $n$ bracketed groups, and each of them contributes (at least) $\frac12$ to the sum. That makes for a total of $\frac n2$, which is the right-hand side in your inequality. – Théophile Dec 04 '15 at 02:10

Discrete Mathematics Proving Question

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