Question on divergent/convergent sums

[u/Ass-Inspector]

if the infinite summation of 1/n diverges, and the infinite summation of 1/n² is the famous π²/6, what is the smallest value k for which the sum converges? Assuming it's not 2 already

[u/Rs3account]

It converges for all k strictly bigger then 1.

Edit: even stronger if we define f(k) = Sum (1/n^k) then f(k) is weldefined for every complex number with a real part strictly larger then 1. And f can be extended to every complex number outside of 1.

[u/productive-man → u/Rs3account]

Can you help me prove something about the non trivial 0s of this function

[u/Rs3account → u/productive-man]

Sure, they all have a real part equal to 1/2.

The proof is easy, I just can't fit it into the reddit response. ;)

[u/Commercial_Handle418 → u/Rs3account]

You could create a link lol 

[u/Rs3account → u/Commercial_Handle418]

It was a joke based on Fermats quote in regards to Fermats last theorem. ;)

[u/Commercial_Handle418 → u/Rs3account]

Oh

[u/axiomus → u/Rs3account]

Does sum of n^-1-lnn [edit: sorry, i meant power to be -(1+1/lnn)] converge or not? i think it diverges but i’m not so sure

[u/snake_case_sucks → u/axiomus]

Converges, ln(n) is increasing.

[u/axiomus → u/snake_case_sucks]

You’re right, i hurriedly made a mistake

[u/No_Yesterday_4260 → u/axiomus]

n^1/ln n = e so the series is just the sum of 1/e n

[u/xnuh → u/axiomus]

For all n>e your exponent is smaller than -2 so the result is smaller than n^-2 so it converges.

The exponent in 1/n^k needs to be smaller than 1 to converge and you put one that gets infinitely big so it pretty clearly converges

[u/axiomus → u/xnuh]

You’re right, i hurriedly made a mistake

[u/Illustrious_Try478 → u/Rs3account]

It's the (non analytically continued) Riemann Zeta function.

[u/Rs3account → u/Illustrious_Try478]

yes, i just didnt think mentioning the zeta function by name would have been usefull. :)

[u/GreaTeacheRopke]

The others are correct

I just want to add, obviously it doesn't matter on Desmos but we generally use p here instead of k. So your question is about "p-series," which is something you can look up if you want to learn more.

[u/Circumpunctilious → u/GreaTeacheRopke]

Thanks—I often just choose a variable (programming habit, probably) and would prefer not to distract people who know better.

[u/GreaTeacheRopke → u/Circumpunctilious]

Of course; I only mentioned it to guide any further research you may do in case you didn't know.

[u/MonitorMinimum4800 → u/Circumpunctilious]

help?? is this guy not clearly ai? he's not even op

edit: well i stand corrected, but still interesting that you responded like you were op

[u/Circumpunctilious → u/MonitorMinimum4800]

!isbot <Circumpunctilious>

ETA: Hm, didn’t work. Don’t know what to tell you, don’t want to spam the comments with other bot / reputation checkers. My stuff isn’t hidden, feel free to look around.

[u/tedecristal]

There's no smallest value.

Any k>1 will converge

[u/Baconboi212121 → u/tedecristal]

Is that for real k? or just integer k?

[u/TheDeadlySoldier → u/[deleted]]

Condensation test is easier no?

[u/EdmundTheInsulter]

My maths lecturer in 1985, he speculated about using a really fast computer to sum the first 10¹⁰ terms of 1/n, saying it wouldn't have got far to infinity, as if some mystical calculation a super computer may have done in an appreciable amount of time.

[u/noonagon]

There isn't a smallest k where it converges, it converges at all k above 1

[u/HelicopterLegal3069]

There is no smallest value of k where the series converges!