For what values of p and q does a series converge?

  • #1
clandarkfire
31
0

Homework Statement


For what values of p and q does [tex]\sum\limits_{n=2}^\infty \frac{1}{n^q\ln(n)^q}[/tex] converge?



The Attempt at a Solution


I've tried a couple of tests, but given that there are two variables (p and q), I'm not really sure how to proceed. My hunch is that I have to use the integral test, which says that the series will converge if [tex]\int^{\infty}_2 \frac{1}{n^q\ln(n)^q}\,dn[/tex] converges, but this isn't something I can integrate nicely, so I'm not quite sure.

Could someone point me in the right direction?
 

Answers and Replies

  • #2
jeppetrost
88
1
I think you've miss typed somewhere. Your sum only has a q, not a p.
 
  • #3
STEMucator
Homework Helper
2,075
140

Homework Statement


For what values of p and q does [tex]\sum\limits_{n=2}^\infty \frac{1}{n^q\ln(n)^q}[/tex] converge?



The Attempt at a Solution


I've tried a couple of tests, but given that there are two variables (p and q), I'm not really sure how to proceed. My hunch is that I have to use the integral test, which says that the series will converge if [tex]\int^{\infty}_2 \frac{1}{n^q\ln(n)^q}\,dn[/tex] converges, but this isn't something I can integrate nicely, so I'm not quite sure.

Could someone point me in the right direction?

Hint : Try considering what's going on around ##q<0##. Try ##q=-1## for example.

How about when ##0 \le q \le 1##?

What can you conclude?
 
  • #4
clandarkfire
31
0
Oh, Sorry! It's actually
For what values of p and q does [tex]\sum\limits_{n=2}^\infty \frac{1}{n^p\ln(n)^q}[/tex] converge?
 
  • #5
STEMucator
Homework Helper
2,075
140
Oh, Sorry! It's actually
For what values of p and q does [tex]\sum\limits_{n=2}^\infty \frac{1}{n^p\ln(n)^q}[/tex] converge?

It's all good. I wish it was just ##q##, would've made it a bit easier.

As for ##p## and ##q## though. Do you know anything about the comparison test?
 
  • #6
clandarkfire
31
0
Yeah! I couldn't think of anything useful to compare it to, though.
 
  • #7
pasmith
Homework Helper
2,269
874
Hint: if [itex]q > 0[/itex] and [itex]\ln n > 1[/itex], then [itex]\displaystyle\frac{1}{n^p (\ln n)^q} < \frac{1}{n^p}[/itex].

Similarly, if [itex]p > 1[/itex] and [itex]n > 1[/itex] then [itex]\displaystyle\frac{1}{n^p (\ln n)^q} <
\frac{1}{n (\ln n)^q}[/itex].

Can you see how to integrate [itex]\displaystyle\int_2^\infty \frac{1}{x (\ln x)^q}\,\mathrm{d}x[/itex] by substitution?
 
  • #8
STEMucator
Homework Helper
2,075
140
Yeah! I couldn't think of anything useful to compare it to, though.

Alright, I'll help you run through one case since there's a couple cases to consider.

The series at hand : ##\sum\limits_{n=2}^\infty \frac{1}{n^p\ln(n)^q}##

Ask yourself, which term in the denominator of your series is your series more dependent on? That is, which function grows faster? ##n## or ##ln(n)##? Forget about ##p## and ##q## for a moment.

Without too much thought it's easy to see ##n>ln(n)## for sufficiently large ##n## independent of ##p## and ##q## and you can use that to your advantage when comparing.

##\sum\limits_{n=2}^\infty \frac{1}{n^p\ln(n)^q} \le \sum\limits_{n=2}^\infty \frac{1}{n^p}##

So for all values of ##q##, the convergence of the series on the left depends on the convergence of the series on the right. What do you know about the series on the right? What can you conclude from all this? This will give you your first case.
 

Suggested for: For what values of p and q does a series converge?

Replies
3
Views
23K
Replies
3
Views
4K
Replies
2
Views
13K
Replies
3
Views
2K
Replies
3
Views
9K
Replies
1
Views
2K
  • Last Post
Replies
2
Views
2K
  • Last Post
Replies
3
Views
10K
Replies
9
Views
1K
Replies
42
Views
6K
Top