TITLE: Let $1=n_0<n_1<\ldots$ be an increasing sequence of positive integers. True/False: $\sum_{i=1}^\infty\frac{n_{i+1}-n_i}{n_{i+1}}$ diverges to $\infty$
QUESTION [6 upvotes]: Let $1=n_0<n_1<n_2<\ldots$ be an increasing sequence of positive integers. Is it true that $\displaystyle\sum_{i=0}^{\infty} \frac{n_{i+1}-n_i}{n_{i+1}} $ diverges to $+\infty?$
For example, if $n_1=2,n_2=3,n_3=53,n_4=54,$ then the first few terms are $\frac{1}{2}, \frac{1}{3}, \frac{50}{53}, \frac{1}{54}.$ Since $\frac{50}{53}<\displaystyle\sum_{k=4}^9 \frac{1}{k},$ this makes me think it might be possible to come up with a counter-exmaple.

REPLY [4 votes]: The series diverges. Hint: for a fixed $k$ and $\ell$ big enough we have $$\frac{n_{k+1}-n_k}{n_{k+1}}+\frac{n_{k+2}-n_{k+1}}{n_{k+2}}+\ldots+\frac{n_{\ell}-n_{\ell-1}}{n_{\ell}}>\frac{n_{k+1}-n_k}{n_{\ell}}+\frac{n_{k+2}-n_{k+1}}{n_{\ell}}+\ldots+\frac{n_{\ell}-n_{\ell-1}}{n_{\ell}} = \frac{n_\ell-n_k}{n_\ell}\ge \frac 12$$