This morning I have explained the monotonous sequences $\{a_n\}$ for my students, but I have had the difficulty of write some simple examples of not-decreasing $a_{n+1}/a_n\geq 1$ and not-increasing sequences $a_{n+1}/a_n\leq 1$.

How to quickly create examples of not decreasing or not increasing monotonous sequences? Is there a specific criterion or method for to create examples?

  • $\begingroup$ $\sin(n)$ is not increasing and not decreasing. $\endgroup$ – hamam_Abdallah Sep 25 at 20:24
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    $\begingroup$ Are you specifically looking for sequences that are monotonic but not strictly monotonic? $\endgroup$ – eyeballfrog Sep 25 at 20:33
  • $\begingroup$ @eyeballfrog Yes, exactly. Thank you very much for to have understood my question. Please, can you edit my bad technical english language? $\endgroup$ – Sebastiano Sep 25 at 20:35
  • $\begingroup$ Take a sequence ${a_n}$ of non-negative terms with some strings of 0's in it. Then ${s_n}$ = the sum of the first n terms of ${a_n}$ will be monotonic but not strictly so. $\endgroup$ – Paul Sep 25 at 20:47
  • $\begingroup$ How about $a_n=\lfloor\sqrt n\rfloor$? -- Btw, $a_{n+1}/a_n\le 1$ would also allow any sequence with alternating signs $\endgroup$ – Hagen von Eitzen Sep 25 at 20:47

Some easy examples of nondecreasing monotonic sequences

  • The constant sequence: $$1,1,1,1,1,1,1...$$ Explicit forumula $a_n = 1$.
  • The integers repeated once each: $$1,1,2,2,3,3,4,4,...$$ Explicit formula: $a_n = \lfloor (n+1)/2\rfloor$
  • The partial decimal expansions of $1/27$: $$0, 0.03, 0.037, 0.037, 0.03703, ...$$ Explicit formula: $a_n = 10^{-n}\lfloor10^n/27\rfloor$
  • The Fibonacci sequence, where the first two terms are $1$ and the remaining terms are the sum of the two previous: $$1, 1, 2, 3, 5, 8, 13, 21, 34, 55 ...$$ Explicit formula: $a_n = \lfloor \phi^n/\sqrt{5}\rceil$, where $\phi = (1+\sqrt{5})/2$ is the golden ratio and $\lfloor x\rceil$ is the nearest integer function.
  • The number of primes less than $n$: $$0, 1, 2, 2, 3, 3, 4, 4, 4, 4, 5, 5, 6, 6, 6, ...$$ Explicit formula: I'd just stick with the primary definition for this one, as it's simple enough to understand why it's not strictly increasing.*

For nonincreasing monotonic sequences, the reciprocals of the above sequences as well as their negations work. I'm having trouble thinking of natural nonincreasing monotonic sequence, though.

*If you must know, it's $a_n =\lceil R(n) - \sum_\rho R(n^\rho)\rceil$, where $R$ is the Riemann $R$ function and the sum runs over all complex roots of the Riemann zeta function.

  • $\begingroup$ Thank you very much. Can you add also nonincreasing examples and the general term of the sequences? I have had the same problem this morning. The students want everything right now. Always thank you and of course I appreciate your answer. $\endgroup$ – Sebastiano Sep 25 at 21:02
  • $\begingroup$ @Sebastiano The general terms of the Fibonacci sequence and especially the prime counting function may be beyond high school algebra, so those sequences should just be explained. The others are $1$, $2\lfloor n/2\rfloor$, and $\lfloor 10^n/27\rfloor/10^n$. $\endgroup$ – eyeballfrog Sep 25 at 21:05
  • $\begingroup$ I assure you with all my heart that Italian high school is not what you think. You can't even imagine my effort, my effort and my concern about elementary algebraic passages. $\endgroup$ – Sebastiano Sep 25 at 21:10
  • $\begingroup$ @Sebastiano Slight correction: the general term of the second sequence is $\lfloor (n+1)/2\rfloor$. $\endgroup$ – eyeballfrog Sep 25 at 21:13
  • $\begingroup$ Can you edit your accepted answer putting the general term near? Hence the examples born to experience :o). Thank you very much. $\endgroup$ – Sebastiano Sep 25 at 21:15

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