Number series convergence/divergence problem

[strange fiber]

my calculus book has the problem to figure out if "Sum of 1 / (ln(n)^3) between n = 2 to infinity" converges or diverges, from my searching for an answer it seems to be a part of complex analysis but idk if there is a method in real analysis to figure this out, I feel like I'm missing something

[crude crescent]

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[vocal grove]

hi

[strange fiber]

hello

[vocal grove]

compare it to some other series

[strange fiber]

i was doing a limit comparison with 1/n but didn't go anywhere i guess i just gotta get better at recognizing which sum supposedly is larger

so i kinda get stuck at not knowing if the sum of 1 series is larger than the other

[red horizon]

you can upper and lower bound it

maybe not upper

definetely lower

observe the behavior of 1/(lnx)^3 as x->infinity

[vocal grove]

1/n will work

[strange fiber]

got a bit weird because i tried the divergence test which gave zero, but i guess that answer is simply inconclusive rather than this series doesn't diverge

[vocal grove]

for any $c\in\bR$, $n>\ln^c(n)$ eventually

[fallow galleon]

fr

[strange fiber]

and this would imply that 1/n is < 1/ln(n)^3 i assume

and since 1/n diverges the series diverges as well

[vocal grove]

of course

the core is actually proving the claim i gave to you

[strange fiber]

yeah, we haven't done many proofs yet

so idk lol

[vocal grove]

try it atleast

just try the c=3 case atleast

cuz thats the only one u care about

[vocal grove]

btw, this isnt even necessary but it's the first method which'd click to you

can you think of some other divergent series, very similar to this and proved very easily too

[strange fiber]

n^2 etc

[vocal grove]

nah nah

kn

[strange fiber]

fair enough

just factoring out the constants but will still make the series larger or smaller depending on k

[vocal grove]

bruh

i mean like

try finding some $k>0$ so that $kn\ge\ln^3(n)$

it's easy

[fallow galleon]

fr

[vocal grove]

just observe that ||lnt=<t for t>0||

[strange fiber]

and just plug in any n to get k after dividing both sides by n

[vocal grove]

nah nah

that's not a proof.

[strange fiber]

we haven't really done proofs idk really where to start

or at least in a formalized way

[vocal grove]

start here

think about what u can do

[cosmic stump]

@strange fiber

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[strange fiber]

+close

[cosmic stump]

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