bound of this sequence

[tepid delta]

what’s your question?

[celest forge]

bound of the sequence

[tepid delta]

what do you think?

consider the absolute value of it first

[celest forge]

yes i tried using the absolute value but it isnt the way we did it at school

we took the highest member

[tepid delta]

how did you do it at school?

[celest forge]

and based on that

we would find how it is

[tepid delta]

well that doesn’t seem like a clear approach

highest member?

[celest forge]

yes

[tepid delta]

you mean the first term?

[celest forge]

yes

[tepid delta]

i see

[celest forge]

my english is not good

but if anything

[tepid delta]

so this sequence is
-1, 1/2, -1/3, 1/4, -1/5, …

in the limit we approach zero yes

[celest forge]

can you tell me a rule that can help me find the bound of sequences

[celest forge]

yes i tried approaching like this

[tepid delta]

generally you have to work on a case by case basis but generally you look at a few things like the first few terms and the limit of the sequence

if it exists

determining if a sequence is monotonic is of course useful

in this case it’s not but the terms decrease in absolute value

[celest forge]

so its the best if j check first if its monotonic

[tepid delta]

usually a good strategy

[celest forge]

and then

can i use the absolute value

[tepid delta]

if it’s monotonically decreasing then it’s bounded above by the first term and bounded below by the limit of the sequence (if it exists) or else it isn’t bounded below

a similar statement can be formed for monotonically increasing

[tepid delta]

here you can and it’s useful because the absolute value is monotonically decreasing

[celest forge]

so

[tepid delta]

here it’s actually

[celest forge]

its based on what kind a sequence it is

[tepid delta]

$-1 \leq a_n \leq \frac{1}{2}$

[thorn isle]

knief

[celest forge]

so its bounded from above and below

[tepid delta]

yea

[celest forge]

at school we would find

the first term

and then by logic i found the second one

1 and ½

but its not a good aproach?

approach*

[tepid delta]

i’m not sure wdym

[tepid delta]

^

[celest forge]

so a1 is 1

[tepid delta]

well -1

[celest forge]

and then i found the second term

yes

the second was ½

[tepid delta]

mhm

[celest forge]

the sequence would circle around 0

[tepid delta]

since the terms decrease in absolute value those are your bounds

[celest forge]

there wasnt anything below -1 and above ½

[tepid delta]

you can show this by considering the subsequences consisting of negative terms and of positive terms separately

[celest forge]

so i thought those are the bounds

[celest forge]

ah

okay

thank you for your help 🙏

[tepid delta]

you’d be correct

no worries

[celest forge]

have a nice rest of your day ☺️

[tepid delta]

you too