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Can I infer the convergence of a monotonically increasing sequence with first term a>0 using the fact that the ratio of consecutive terms tends to 1?

Thanks in advance.

Panteleimon Tassopoulos
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1 Answers1

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For a positive, monotonically increasing sequence $(x_n)$ of real numbers, consider the statements $(\mathbf{A}),(\mathbf{B})$ given by

$\qquad (\mathbf{A})\qquad(x_n)\;$ converges.

$\qquad (\mathbf{B})\qquad{\displaystyle{\lim_{n\to\infty} \frac{x_{n+1}}{x_n}=1}}$.

The implication$\;(\mathbf{A}){\implies}(\mathbf{B})\;$holds since if $(x_n)$ converges to a limit, $L$ say, then $L > 0$ and $$\lim_{n\to\infty} \frac{x_{n+1}}{x_n}=\frac{L}{L}=1$$ However the converse implication$\;(\mathbf{B}){\implies}(\mathbf{A})\;$doesn't hold. As a simple counterexample, for the sequence $1,2,3,...$ we have $$ \lim_{n\to\infty} \frac{x_{n+1}}{x_n} =\lim_{n\to\infty} \frac{n+1}{n} =\lim_{n\to\infty} 1+\frac{1}{n} =1+0 =1 $$ but $(x_n)$ approaches infinity, so doesn't converge.

quasi
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  • This does not provide an answer to the question. To critique or request clarification from an author, leave a comment below their post. - From Review – Thomas Lesgourgues Apr 20 '20 at 22:14
  • @Thomas Lesgourgues: Doesn't the simple counterexample I gave show that one can't infer convergence from the OP's specified hypothesis? – quasi Apr 20 '20 at 22:31
  • It does indeed, but I wouldn't qualify this a quality response. It would be a nice comment. Some additional details are required (in my opinion) to make this a valid answer. Others might disagree and I won't mind if your answer is validated. – Thomas Lesgourgues Apr 20 '20 at 22:39
  • In this case, I think the simplest counterexample is the right answer. It illustrates that a positive, increasing sequence can approach infinity and still have ratios of consecutive terms approaching $1$. – quasi Apr 20 '20 at 22:42
  • I agree that it could be, especially for the OP's need. However, I would think that an actual answer would include missing hypothesis for example : is an additional hypothesis necessary / sufficient to ensure convergency ? Or is the OP's hypothesis way too weak ? I don't think that the OP will gain much from just this line. – Thomas Lesgourgues Apr 20 '20 at 22:46
  • After the OP looks at my answer, if the OP wants to adjust the hypothesis, fine, I'll adjust my answer, but I suspect the OP's intended question is just what was posted. – quasi Apr 20 '20 at 22:50