• Support PF! Buy your school textbooks, materials and every day products Here!

Finding the value of p so that the series converges

  • Thread starter Dell
  • Start date
  • #1
590
0
find the possible values of p so that the following converges

infinity
[tex]\sum[/tex]1/(ln(n)*np)
n=2

what i thought of doing was integrating to find the values of p such that the integral will give me an answer not infinity.

[tex]\int[/tex]dx/(ln(x)*xp) from 2-infnity

i thought of using substitution as
t=ln(x)
x=et
dt=dx/x

[tex]\int[/tex]dt/(t*et(p-1)) from ln2-infinty


but i have no idea how to continue from here, i think that if p<=1 then my integral will diverge since lim(t->inf) will not be 0, but i am not sure, i know that with a series if lim(n->inf) is not 0 then the series diverges, is this true for integrals as well,
even so, if i know that p<=1 diverges, this does not automatically mean that anything else converges, since lim(n->inf)An=0 doesnt mean necessarily converges, how do i find the values for p where i KNOW the integral diverges??
 

Answers and Replies

  • #2
Your argument works for p <= 1.

To do arguments for p > 1, use a comparison test.. Show that each term (n>2) is less then
1/n^p.
 
  • #3
590
0
so do i say:
for any n>2 my series is smaller than 1/n^p, and if p>1 1/n^p converge, therefore mine must also diverge, as for p<1 i already proved when i said lim(n->inf) not 0, so my series must diverge, therefore the series only converges when p>1,

do you see a better way of solving this?
 

Related Threads for: Finding the value of p so that the series converges

Replies
11
Views
2K
Replies
15
Views
477
Replies
3
Views
21K
Replies
1
Views
2K
Replies
1
Views
13K
Replies
9
Views
902
  • Last Post
Replies
9
Views
1K
Replies
5
Views
2K
Top