Question

If a sequence is monotonic , it must be bounded and why ?

Answer 1

I don't know much about analysis but isn't the sequence \(a_n=n^2\) monotonic yet unbounded?

Answer 2

For a monotonic sequence to converge, it of course must be bound, because otherwise, it will go into positive or negative infinity (that is, diverge)

Answer 3

Also, a quick important note.

Whenever sequence doesn't converge, series all the more so doesn't converge

Answer 4

Should I give some examples of bound and unbound sequences (for sequence convergence/divergence) ?

Answer 5

yes , please :)

Answer 6

For example \(\large\color{black}{ \displaystyle \left\{ \frac{1}{n} \right\}_{n=1}^{\infty}}\).
In this sequence, you are always decreasing (because `1/a > 1/(a+1)` ) and this way it is monotonic.
it is at most 1 (for n=1), and it converges to zero

(you can induct that, or you know that \(\large\color{black}{ \displaystyle \lim_{n\rightarrow \infty}\left[ \frac{1}{n} \right]=0}\) )

Answer 7

(this is sequence, but if you add the terms - the series, then this would diverge... there are so many proves for divergence [!!].... the series of 1/n is known as the HARMONIC SERIES.

Answer 8

but, sequence converges.

Answer 9

it is monotonic and bound by 1 and 0

Answer 10

by [1,0)

Answer 11

would you agree that any arithmetic sequence (asides from {3, 3, 3, 3, 3 ...} or any {a, a, a, a, a ...} ?

(why? well, it is not bound, so you can be adding or subtracting depending on the pattern forever... this will for sure get you to \(\pm\) infinity.

Answer 12

arithmetic examples

\(\large\color{black}{ \displaystyle \left\{ 10- n \right\}_{n=0}^{\infty}}\)

\(\large\color{black}{ \displaystyle \left\{ n+1 \right\}_{n=0}^{\infty}}\)

\(\large\color{black}{ \displaystyle \left\{ A_n \right\}_{n=0}^{\infty}}\) for any arithmetic sequence...



for all of them, they would diverge. Would you agree to this, ro you have a question about this particular post?

Answer 13

now, geometric examples

\(\large\color{black}{ \displaystyle \left\{ 2n \right\}_{n=0}^{\infty}}\)
you can get as big as you want, b/c it is not bound.

Answer 14

also,

\(\large\color{black}{ \displaystyle \left\{ 1-3n \right\}_{n=0}^{\infty}}\)
(can get as small as you wanty, and will approach -\(\infty\) )

Answer 15

I have one question about arithmetic sequence {a, a, a, a, a ...} , is it a monotonic sequence ?

Answer 16

these last 2 are arithmetic, excuse me...

here are some geometric sequences though...

\(\large\color{black}{ \displaystyle \left\{ 2^n \right\}_{n=0}^{\infty}}\)

\(\large\color{black}{ \displaystyle \left\{ a^n \right\}_{n=0}^{\infty}}\) for \(a>1\)

those will grow as large as possible, because there is no boundary.

((( yes, and {a,a,a,a,a} is NOT monotonic, because it is:
~ not decreasing, because \(a_{n}>a_{n+1}\) is not true.
~ not increasing, because \(a_{n}<a_{n+1}\) is not true.

3=3 , but not 3>3 or 3<3 )))

Answer 17

also, {a,a,a,a,a} is both arithmetic and geometric for all real number a, except a=0.

Answer 18

if you have more questions, please ask

Answer 19

as a result , can we say both arithmetic and geometric sequences . can be monotonic or not monotonic , but always diverge?

then every monotonic sequence , must be bounded?

Answer 20

Arithmetic sequence is always monotonic (and will diverge),
unless it is `{a, a, a, a, a ...}` which is not monotonic, but converges to "a".
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Geometric sequence is convergent if it is bound and monotonic.
But, if geometric sequence is not monotonic, such as involving (-1)\(^{\large \rm n}\), or a trig function, because the \(\large\color{slate}{\displaystyle\lim_{n \rightarrow ~\infty}A_n}\) will diverge to \(\pm \infty\) or in either case it will alternate between -1 and 1 (which means divergence).

this is because limit has to approach ONE value (not multiple values).

Answer 21

For arithmetic sequence, unless it is {a, a, a, a, a....} , it will diverge because arithmetic sequence is not bound. In fact, never bound, except [a, a, a, a... } , and if we are taking about an infinite sequence of course.

Answer 22

now it becomes clear ..

really ,Thank you very much :)

Answer 23

yw