I suggested merging two questions in Math Mods' office, quoting more of its duplicates here: but they seem to have interesting (and different) solutions to the problem, can they all be merged to one?
So I've been thinking through some test cases. If $a_n = n$ then $\sum_n \frac{a_{n+1} - a_n}{a_n}$ is the harmonic series which diverges. And if $a_n = \sum_{k=1}^n 1/k$ then $\sum_n \frac{a_{n+1} - a_n}{a_n}$ diverges like $\sum_n 1/(n \log n)$. So that got me thinking, if $a_n$ is a strictly i...
Let $(a_n)_{n\ge 1}$ is an increasing sequence of of positive numbers such that $a_n \to \infty$ as $n \to \infty $. Then $\sum_{n=1}^\infty \frac{a_{n+1}-a_n}{a_{n+1}}$ diverges.
Let $(a_n)$ be an unbounded strictly increasing sequence of positive real numbers and let $x_k=\frac{a_{k+1}-a_{k}}{a_{k+1}}$. Then I want to find the correct option (s).
For all $n\geq m, \sum\limits^{n}_{k=m}x_k>1-\frac{m}{n}$
For all $n\geq m, \sum\limits^{n}_{k=m}x_k>\frac{1}{2}$
$\sum\limi...
Let $\{a_n\}_{n=1}^\infty$ be a strictly increasing sequence with positive terms; discuss the convergence of the series $\displaystyle\sum_{n=1}^\infty\frac{a_{n+1}-a_n}{a_n}$.
Let $\{a_n\}_{n=1}^\infty$ be a strictly increasing sequence with positive terms; discuss the convergence of the series $\displaystyle\sum_{n=1}^\infty\frac{a_{n+1}-a_n}{a_n}$.
