The $a_n$'s are integers, positive, and increasing: $0< a_1 < a_2 < \cdots$, the problem asks us to prove that: $$ \sum^{\infty}_{n=1} \frac{a_{n+1}-a_{n}}{a_{n}}=\infty $$ While I have checked this results for several series like $a_n = n$, $a_n = n^2$, $a_n = n^p$, or $a_n = p^n$ type stuff, I don't know how to prove this general result. A hint is appreciated. Thanks dudes!
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5...and dudettes! – Bruno Joyal Nov 13 '13 at 18:49
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1@tony:do not yet have the privilege of posting a comment, so putting this comment as an answer.Did you try out the sum for the situation when a(n+1)/a(n) = 1+(1/n^2)? – Sudhir Nov 14 '13 at 16:46
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1Dear Sudhir: in the question, the $a_n$'s are integers. – Bruno Joyal Nov 14 '13 at 18:27
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Consider a partial sum: $\displaystyle{\large S_{N} \equiv \sum_{n = 1}^{N}{a_{n + 1} - a_{n} \over a_{n}} = \sum_{n = 1}^{N}{a_{n + 1} \over a_{n}} - N > 0}$. – Felix Marin Nov 15 '13 at 04:23
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@FelixMarin And so what? Your comment seems like a misleading hint to me. If it is not, please explain. – Did May 05 '16 at 09:19
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I never thought I'd answer a question asked by a superhero! I would advise Mr. Iron Man to use the following well-known theorem:
Let $\{x_i\}$ be a sequence of positive real numbers. Then the product
$$\prod_{i=1}^\infty (1+x_i)$$
converges if and only if the series
$$\sum_{i=1}^\infty x_i$$
converges.
In the present case case, notice that
$$1+\frac{a_{i+1}-a_i}{a_i} = \frac{a_{i+1}}{a_i}$$
so the partial products of the infinite product telescope, to give $a_{n+1}/a_1$, which tends to $+\infty$ by assumption. Therefore, the series $\sum \frac{a_{i+1}-a_i}{a_i}$ diverges.
Remark Your series is analogous to the integral $$\int_0^\infty df/f$$ where $f$ is a positive function. Of course, this integral equals $\varinjlim_{x \to \infty} \log (f(x)/f(0))$, which is $+ \infty$ if $f \to \infty$.
Bruno Joyal
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$\dfrac{a_{n+1}-a_n}{a_n} = \dfrac{\Delta a_n}{a_n}$, so it does look like $\dfrac{df}{f}$, but if I had answered this the answer would not have been so efficient. – Michael Hardy Nov 13 '13 at 18:54
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1@MichaelHardy What do you mean? Thanks for the edit btw. You were right about the French! – Bruno Joyal Nov 13 '13 at 18:55
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@BrunoJoyal, ok can you please justify the move from $\dfrac{f(x+1)-f(x)}{f(x)}$ to $\dfrac{df}{f}$ – toufik_kh.17 Apr 11 '14 at 22:25
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3@toufik_kh.17 The analogy here is that the forward difference operator on sequences is analogous to the derivative. It's just an analogy, don't try to read too much into it... – Bruno Joyal Apr 11 '14 at 22:46
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We have $$\sum_{i=1}^{n} \frac{a_{i+1}-{a_i}}{a_i} \geq \sum_{i=1}^{n}\int_{a_i}^{a_{i+1}}\frac{1}{x}\rm{d}x=\ln\left(\frac{a_{n+1}}{a_1} \right )$$ taking $n\rightarrow \infty$ we get the result.
clark
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The key is that there is a constant $c > 0$ such that $a_{n+1} \ge a_n+c$. – marty cohen May 02 '16 at 22:19