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Give an example of a sequence which is neither increasing after a while, nor decreasing after a while, yet which converges to 1.

My solution: $1.01,\ .99,\ 1.001,\ .999,\ 1.0001,\ .9999,\ \text{etc}\dots$

Does that satisfy all the conditions? Also, judging by the instructions, do you think I would have to define that sequence? In which case, I could do $\{x_n\} = 1 + .01^n$ for odd $n$ and $1 - .01^n$ for even $n$ (which would change the sequence, but just increases the rate at which it approaches $1$).

The definition of an increasing sequence used is the next term being bigger than OR equal to the preceding term. And dually for decreasing.

Remy
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7 Answers7

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I think defining it the way you did is fine, but you could also use

$$s_n = 1+\left(-\frac{1}{10}\right)^n$$

which is a handy little trick to express alternating sequences nicely.

Carl Schildkraut
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    I think it just has to be a sequence that doesn't increase or decrease the whole time (1.1, 1.01, 1.001 or .9,.99,.999) – Remy Sep 20 '16 at 02:58
  • But I could be reading the instructions incorrectly. – Remy Sep 20 '16 at 02:59
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    Oh, I just realized. 1,1,1,1,1 is both increasing and decreasing. Since the definition given by my prof for increasing is the next term being bigger than or equal to the preceding term. – Remy Sep 20 '16 at 03:32
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    @Brick : The instruction "neither increasing ..., nor decreasing ..." means that there is no finite initial segment after which the sequence only decreases or only increases. No tail of this answer's sequence only decreases or only increases. – Eric Towers Sep 20 '16 at 05:30
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    You can make the expression lower with another minus in the exponent: $$1+(-10)^{-n}$$ – CiaPan Sep 20 '16 at 10:38
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Your solution works.

I like $1+\dfrac{\sin n}{n}$. But it's possible I'm weird.

Eric Towers
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  • I was going to post this too, but someone other than op edited in criteria for an increasing/decreasing function, so depending on how strict the initial requirements are, this may or may not work. – Burnsba Sep 20 '16 at 12:17
  • @Burnsba : Since no two adjacent terms of this sequence are equal, the non-strictness of "increasing" and "decreasing" does not impact its validity. (We would look for $k \in \Bbb{Z}$ such that $\dfrac{(2k+1)\pi}{2} \pm \dfrac{1}{2}$ are both integers, but this would require that the transcendental number $\dfrac{(2k+1)\pi}{2}$ is rational, which is absurd.) – Eric Towers Sep 20 '16 at 12:51
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    @Burnsba : The instruction "neither increasing ..., nor decreasing ..." means that there is no finite initial segment after which the sequence only decreases or only increases. No tail of this answer's sequence only decreases or only increases. If we were to take your interpretation, no solution is possible. – Eric Towers Sep 20 '16 at 13:05
  • This is what I was going to suggest. – Brian Sep 20 '16 at 20:11
  • I like the intuitive appeal, but it is admittedly kind of subtle to get the exact proof for this one -- a bit of a rabbit hole dive – MichaelChirico Sep 21 '16 at 02:29
  • @Burnsba FYI the information I edited in was literally given by OP in a comment, in fact even twice in comments. Plus as explained by others it's irrelevant for this example. The only example to which it is really relevant is that of an (eventually) constant sequence. – quid Sep 21 '16 at 11:25
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EDIT: As the comments note, the original formulation of the question was not specific about whether the inequalities in the definitions were strict. With the additional specification now that "increasing" and "decreasing" allow for equality, this answer of course doesn't fit anymore.

You could just take the sequence 1, 1, 1, 1, 1, ... .

Brick
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    Oh, true... I didn't think of that. I guess that would work since the definition of an increasing sequence is the next term being bigger than OR equal to the preceding term. – Remy Sep 19 '16 at 23:41
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    That technically works, although the question is rather vague about strict vs. non-strict increasing and decreasing. – Carl Schildkraut Sep 19 '16 at 23:41
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    I’m sure some people do take “strictly in/decreasing” as their default reading of “in/decreasing sequence”, in which context this answer is very nice. But it’s much more common in my experience for “in/decreasing sequence” to mean the notion with ≤ or ≥, in which case this answer doesn’t satisfy it. – Peter LeFanu Lumsdaine Sep 20 '16 at 07:41
  • @RemyM " I guess that would work since the definition of an increasing sequence is the next term being bigger than OR equal to the preceding term." But this sequence has the property that the next term is "bigger than OR equal to the preceding term." Thus it is an increasing sequence and not an example of what you are looking for. – quid Sep 20 '16 at 11:46
  • This is answer was from the start highly misleading. The terminology is not uniform but given the context it was pretty clear that the constant sequence would count as increasing (and decreasing), by now confirmed by OP, making this is answer false. I do not get how you can insist on your example after OP recalled the definition. – quid Sep 20 '16 at 11:50
  • @CarlSchildkraut Slightly ironically the subject was clarified in the comment right before yours. But perhaps it was so close in time it was not there yet when you wrote your comment. Given that clarification I assume you will agree the example does in fact not work. – quid Sep 20 '16 at 11:58
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    @quid I've always seen increasing to mean strictly increasing, but if that's not the usual meaning then I guess this answer wouldn't work. – Carl Schildkraut Sep 20 '16 at 13:59
  • @quid I updated the answer to reflect the evolving formulation of the question. The first comment by the OP under this answer was ambiguous since he accepts the answer but then contradicts himself by adding that his definition of increasing/decreasing allows for equality. His later comments on other answers do seem to point to allowing for equality though. I'm with Carl though that I'm used to seeing "increasing" mean "strictly increasing" by default. – Brick Sep 20 '16 at 14:03
  • @Brick it was not ambiguous. They just got confused regarding its consequences. Either way, as a rule it is not a good idea to give examples that depend on subtle nuance of a definition without making this explicit, especially not when easy examples are available that work regardless. But thanks for the edit. – quid Sep 20 '16 at 14:11
  • @CarlSchildkraut whatever the usual meaning, OP specified what they meant. (I think it is a language/cultural/regional thing; for instance is French "croissant" and "decroissant" will usually include the constant sequences, so that the more idiomatic translation of "croissant" would be "non-decreasing" and of "decroissant" "non-increasing" but somebody asking the question above is unlikely to be aware of those differences in usage and go be by the default translation.) – quid Sep 20 '16 at 14:16
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I think it is possible to use a well known Fibonacci sequence property:

$$\lim_{n\to\infty}\frac{F_{n}}{F_{n-1}} = \varphi$$

where $\varphi$ is the golden ratio, and also it is well known that each term of the sequence, i.e. let us call it $a_n=\frac{F_{n}}{F_{n-1}}$, oscillates between a value over and under $\varphi$. For that reason the sequence $$\left\{n \gt 2:a_n=\frac{F_{n}}{F_{n-1}\cdot \varphi}\right\}$$ will be neither increasing after a while, nor decreasing after a while, and will converge to $1$.

CiaPan
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iadvd
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    You can turn this into a rational sequence by dividing two consecutive terms: $b_n = F_{n-1} F_{n+1} / F_n^2$. – Erick Wong Sep 20 '16 at 10:44
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    I've made a little fix into your post. If you prefix lim with a backslash you convert it into a LaTeX symbol, which is properly rendered in upright font: \lim → $\lim$, and has a limit condition properly put underneath it, instead of italics, which looks like a $l\cdot i\cdot m$ multiplication: lim → $lim$ with a condition appended to 'em' . – CiaPan Sep 20 '16 at 10:46
  • @CiaPan thank you for that change! – iadvd Sep 20 '16 at 11:05
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Yes, your sequence satisfies all required conditions.

Another example sequence: $$a_n = \begin{cases} 1+\frac 1n & \text{if } \log_{10}n \in \mathbb N \cup \{0\} \\ 1 & \text{otherwise} \end{cases}$$ has terms: $$\begin{align} a_1 & = 2 \\ a_{10} & = 1.1 \\ a_{100} & = 1.01 \\ a_{1000} & = 1.001 \\ a_{10000} & = 1.0001 \\ \ldots \end{align}$$ and all remaining terms equal $1$.

CiaPan
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It's so easy to find such a series, for example: $$a_n=q^n+1\ ,\ -1<q<0$$

76david76
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2

Continued fraction representations of irrational numbers can be represented as an infinite sequence of rational partial convergents which oscillate above and below the number being represented.

Thus if we take $a_n$ to be the to be the nth partial convergent of any irrational number divided by that number, it will converge to 1 in an oscillatory manner.

$$ a_n = \frac{[\zeta]_n}{\zeta}, \zeta \in \mathbb{I} $$

For $\zeta = \sqrt{2}$, we have

$$ a_0 = \frac{1}{\sqrt{2}} \approx 0.707\\ a_1 = \frac{3}{2\sqrt{2}} \approx 1.061\\ a_2 = \frac{7}{5\sqrt{2}} \approx 0.990\\ a_3 = \frac{17}{12\sqrt{2}} \approx 1.002\\ ... $$

James
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