Is it true that for every strictly increasing sequence $ (n\_k)\_{k\ge 1} $ of natural numbers, the sequence $ \frac{n\_k - n\_{k−1}}{n\_k} $ is never summable? 

A proof is easy in the case when $ \limsup\_{k\to\infty} \frac{n\_{k-1}}{n\_k} < 1 $ simply because then the individual terms don't converge to 0, but when the $ \limsup $ equals 1, how does one proceed?