Timeline for $\sum\limits_{n=1}^{\infty}(-1)^n ((\frac{a_{n+1}}{a_{n}})^2-1)$ converges.Does $\sum\limits_{n=1}^{\infty}(-1)^n (\frac{a_{n+1}}{a_{n}}-1)$ converge?
Current License: CC BY-SA 3.0
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| May 18, 2018 at 20:43 | vote | accept | CommunityBot | ||
| Aug 15, 2017 at 9:25 | comment | added | Wadim Zudilin | I have already given a solution in the yesterday comment: take, e.g., $y_{2n-1}=0$ (that is, $a_{2n}=a_{2n-1}$) and $y_{2n}=(-1)^n/\sqrt{n}$ (that is, $(a_{2n+1}/a_{2n})^2=1+(-1)^n/\sqrt{n}$). Then $\sum_{n=1}^\infty y_n^k$ converges for any $k\ge1$ different from 2, in particular, for $k=1$ and 3, and diverges for $k=2$. Thus $\sum_{n=1}^\infty x_n$ diverges. The connection between $x_n$ and $y_n$ (was that your question?) comes from solving $x(2+x)=y$. | |
| Aug 15, 2017 at 2:09 | answer | added | Terry Tao | timeline score: 5 | |
| Aug 15, 2017 at 1:49 | answer | added | Christian Remling | timeline score: 3 | |
| S Aug 14, 2017 at 15:42 | history | suggested | user113386 | CC BY-SA 3.0 |
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| Aug 14, 2017 at 15:16 | review | Suggested edits | |||
| S Aug 14, 2017 at 15:42 | |||||
| S Aug 14, 2017 at 15:08 | history | suggested | user113386 | CC BY-SA 3.0 |
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| Aug 14, 2017 at 14:51 | review | Suggested edits | |||
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| Aug 14, 2017 at 13:57 | history | edited | user113386 | CC BY-SA 3.0 |
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| Aug 14, 2017 at 13:56 | comment | added | user113386 | @WadimZudilin: Cannot fetch last $y_n^2$. Where did 2 come from? | |
| Aug 14, 2017 at 13:46 | comment | added | Wadim Zudilin | The summation over $n\ge1$ can be shifted to $n\ge n_0$ without changing the problem; in particular, $a_n>0$ for all $n$ can be dropped as $a_{n+1}/a_n\to1$ implies that $a_n$ is eventually sign-definite. Denote $x_n=(-1)^n(a_{n+1}/a_n-1)$ and $y_n=(-1)^n((a_{n+1}/a_n)^2-1)$; we have $x_n\to0$ and $y_n\to0$ as (from $a_{n+1}/a_n\to1$), so that $x_n=y_n/2-(-1)^ny_n^2/8+O(y_n^3)$ as $n\to\infty$. Assuming $\sum_{n\ge n_0}y_n$ converges, the divergence of $\sum_{n\ge n_0}x_n$ will potentially require $\sum_{n\ge n_0}(-1)^ny_n^2$ to diverge. | |
| Aug 14, 2017 at 12:14 | history | edited | user113386 | CC BY-SA 3.0 |
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| Aug 14, 2017 at 12:01 | history | edited | user113386 | CC BY-SA 3.0 |
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| Aug 14, 2017 at 11:27 | history | edited | user113386 | CC BY-SA 3.0 |
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| Aug 14, 2017 at 11:22 | history | edited | user113386 | CC BY-SA 3.0 |
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| Aug 14, 2017 at 11:03 | review | First posts | |||
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| Aug 14, 2017 at 11:00 | history | asked | user113386 | CC BY-SA 3.0 |