- #1

- 590

- 0

i need to tell if the following sequence converges or diverges

[tex]\sum[/tex](-1)

(n from 1 to infinity)

what i think i need to do is take the positive version of this series

[tex]\sum[/tex]ln

(n from 1 to infinity)

if the positive series converges than the original must also, if not i can use leibnitz to tell me its behaviour

the problem it the "p", surely {p>0} is not enough information, do i not need to know if p>1 or p<1???

i can integrate [tex]\int[/tex](ln

[tex]\int[/tex]t

=t

...ln

if P>-1 [tex]\sum[/tex]ln

if P<-1 [tex]\sum[/tex]ln

if p=-1 [tex]\sum[/tex]ln

but i am told that {p>0} therefor i am only left with one option

if P>-1 [tex]\sum[/tex]ln

so now i need to check if:

- lim An = 0

- An+1< An

[tex]\sum[/tex](-1)

^{n-1}ln^{p}(n)/n {p>0}(n from 1 to infinity)

what i think i need to do is take the positive version of this series

[tex]\sum[/tex]ln

^{p}(n)/n {p>0}(n from 1 to infinity)

if the positive series converges than the original must also, if not i can use leibnitz to tell me its behaviour

the problem it the "p", surely {p>0} is not enough information, do i not need to know if p>1 or p<1???

i can integrate [tex]\int[/tex](ln

^{p}(x)/x)dx ===> t=lnx dt=dx/x[tex]\int[/tex]t

^{p}dt=t

^{p+1}/(p+1) =......ln

^{p+1}(n)|[tex]^{infinity}_{1}[/tex]if P>-1 [tex]\sum[/tex]ln

^{p}(n)/n divergesif P<-1 [tex]\sum[/tex]ln

^{p}(n)/n convergesif p=-1 [tex]\sum[/tex]ln

^{p}(n)/n divergesbut i am told that {p>0} therefor i am only left with one option

if P>-1 [tex]\sum[/tex]ln

^{p}(n)/n divergesso now i need to check if:

- lim An = 0

- An+1< An

Last edited: