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So I'm trying to figure out a homework problem. We have to prove it using weak induction.

$$\sum_{j=1}^{2^n}\frac{1}{j}\ge1+\frac{n}{2}$$

The base case is easy, and I have my induction hypothesis as the sum from $j=1$ to $m$ of $\frac{1}{j}$ is greater than or equal to $1 + \frac{m}{2}$. But I can't figure out how to get to the $n+1$ step. Help?

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  • $\begingroup$ Welcome to MSE! You can find a tutorial on how to format your posts with MathJax here. It's standard to do so here, and the basics of MathJax are pretty easy to pick up. $\endgroup$ – Robert Howard Feb 21 at 21:50
  • $\begingroup$ I think this is easier if you actually expand the terms of this series, and then group them in terms of what you're adding for each increment of $n$: $\left(\frac11\right) + \left(\frac12\right) + \left(\frac13+\frac14\right)+\left(\frac15+\frac16+\frac17+\frac18\right)+\cdots$. Does that make sense? $\endgroup$ – Brian Tung Feb 21 at 21:53
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Suppose that the claim holds for some integer $n$. Then note that $$ \sum_{j=1}^{2^{n+1}}\frac{1}{j}=\sum_{j=1}^{2^n}\frac{1}{j}+\sum_{j=2^n+1}^{2^{n+1}}\frac{1}{j}\ge 1+\frac{n}{2}+\frac{2^n}{2^{n+1}}=1+\frac{n+1}{2} $$ where we used the induction hypothesis to get the inequality and used $$ \sum_{j=2^n+1}^{2^{n+1}}\frac{1}{j}\geq \frac{2^n}{2^{n+1}} $$ since we are summing $2^n$ terms each of which is at least $1/2^{n+1}$.

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