Sequences and Series

  • Thread starter cathy
  • Start date
  • #1
90
0

Homework Statement




So, I actually have a bunch of these problems and I cannot do any of them. I don't think I'm really understanding it. Here is the question: (one of them) The way I wrote them, a_n means a sub n

For each sequence a_n find a number k such that n^k a_n
has a finite non-zero limit.
(This is of use, because by the limit comparison test the series ∑from 1 to infinity (a_n) and ∑from 1 to infinity (n^-k) both converge or both diverge.)

The question:
a_n= (6+3n)^-7
What does k equal?

Homework Equations


N/A


The Attempt at a Solution


I'm actually 100% lost on this problem. I don't even know where to start :(
Here's my attempt. It may not make any sense because I don't understand this. But here it is:
(6+3n)^-7>0
1/(6+3n)^7>0
1>0
Ans: 1
But that is incorrect.
 

Answers and Replies

  • #2
Dick
Science Advisor
Homework Helper
26,258
619

Homework Statement




So, I actually have a bunch of these problems and I cannot do any of them. I don't think I'm really understanding it. Here is the question: (one of them) The way I wrote them, a_n means a sub n

For each sequence a_n find a number k such that n^k a_n
has a finite non-zero limit.
(This is of use, because by the limit comparison test the series ∑from 1 to infinity (a_n) and ∑from 1 to infinity (n^-k) both converge or both diverge.)

The question:
a_n= (6+3n)^-7
What does k equal?

Homework Equations


N/A


The Attempt at a Solution


I'm actually 100% lost on this problem. I don't even know where to start :(
Here's my attempt. It may not make any sense because I don't understand this. But here it is:
(6+3n)^-7>0
1/(6+3n)^7>0
1>0
Ans: 1
But that is incorrect.
Here's a hint. (6+3n)=n(6/n+3). Try and do something with that.
 
  • #3
90
0
Would you mind explaining the meaning? How would I find k from it?
 
  • #4
Dick
Science Advisor
Homework Helper
26,258
619
Would you mind explaining the meaning? How would I find k from it?
Ok, so (6+3n)^(-7)=(n(6/n+3))^(-7)=n^(-7)*(6/n+3)^(-7). What's the limit n->infinity of (6/n+3)^(-7)?
 
  • #5
90
0
isn't it infinity?
 
  • #6
Dick
Science Advisor
Homework Helper
26,258
619
isn't it infinity?
I don't think so. Why do you? What's limit 6/n as n->infinity?
 
  • #7
90
0
because isnt it the same as saying ((n+3)/6)^7 so as n approaches infinity, it approaches infinity?
 
  • #8
90
0
but how would doing that help me find what k is?
 
  • #9
Dick
Science Advisor
Homework Helper
26,258
619
because isnt it the same as saying ((n+3)/6)^7 so as n approaches infinity, it approaches infinity?
No, it isn't. ((6/n)+3)^(-7)=1/((6/n)+3)^7. It's not 6/(n+3). It's (6/n)+3. They are very different.
 
  • #10
90
0
and using this new limit (6/n)+3, how would it lead me to solve for k?
 
  • #11
Dick
Science Advisor
Homework Helper
26,258
619
and using this new limit (6/n)+3, how would it lead me to solve for k?
I think you should stop asking that until you do the algebra correctly and tell me what limit n->infinity ((6/n)+3) is. Then take a breath and think about it.
 
  • #12
90
0
the limit is 3 because the 6/n goes to 0
 
  • #13
Dick
Science Advisor
Homework Helper
26,258
619
the limit is 3 because the 6/n goes to 0
Ok, so what's limit ((6/n)+3)^(-7). Have you thought about what that might mean for your question of what k is?
 
  • #14
90
0
1/3^7= 1/2187.
I still dont know where k comes into play.
 
  • #15
90
0
Is k 7?
 
  • #16
90
0
Here's a hint. (6+3n)=n(6/n+3). Try and do something with that.
Where did you get the n(6/n+3) here? Why was it necessary to factor out the n?
 
  • #17
Dick
Science Advisor
Homework Helper
26,258
619
Is k 7?
Ok, so (6+3n)^(-7)=(n(6/n+3))^(-7)=n^(-7)*(6/n+3)^(-7). As n->infinity (6/n+3)^(-7)->1/3^7. Does it make sense to you that k=7 works??
 
  • #18
Dick
Science Advisor
Homework Helper
26,258
619
Where did you get the n(6/n+3) here? Why was it necessary to factor out the n?
Because you want the series to look like n^(-k) times something that approaches a nonzero constant. It's a good idea to factor out the n and see what's left.
 
  • #19
90
0
yes it does. so just to recap, i had to factor out an n to create the n^k? what if n were in the demonimator instead?
 
  • #20
Dick
Science Advisor
Homework Helper
26,258
619
yes it does. so just to recap, i had to factor out an n to create the n^k? what if n were in the demonimator instead?
Then you would factor out a 1/n^(k)=n^(-k). It just changes the sign of k. Doesn't it?
 
  • #21
90
0
i have a problem that says 7/(n^3+n) you dont have to solve this, but i just want to know, would i have to factor out n? and i would have 7n^-3?
 
  • #22
Dick
Science Advisor
Homework Helper
26,258
619
i have a problem that says 7/(n^3+n) you dont have to solve this, but i just want to know, would i have to factor out n? and i would have 7n^-3?
That will work. What's the limit of what's left after you factor that out?
 
  • #23
90
0
is it 0?
 
  • #24
Dick
Science Advisor
Homework Helper
26,258
619
is it 0?
Is that a guess? What's left over after you factor 7n^(-3)) out? Mind your algebra.
 
  • #25
90
0
1/n^4
 

Related Threads on Sequences and Series

  • Last Post
Replies
4
Views
2K
  • Last Post
Replies
6
Views
2K
  • Last Post
Replies
2
Views
1K
  • Last Post
Replies
3
Views
941
  • Last Post
Replies
1
Views
1K
  • Last Post
Replies
6
Views
2K
  • Last Post
Replies
4
Views
1K
  • Last Post
Replies
3
Views
2K
  • Last Post
Replies
1
Views
925
  • Last Post
Replies
6
Views
2K
Top