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

Convergence of a Series

  • Thread starter mohabitar
  • Start date
  • #1
140
0
So we have the series from n=1 to infinite of 1/(n^3+n), and we're supposed to use the integral test, which works out and gives the answer of 1/2 ln2, so it converges.

My question is why cant we use the limit test for this series? Or wait can we? If we take the limit, it would be zero, since we would divide each tearm by n^3 and it would come out to zero.

So basically the book says use the integral test, and thats cool and all, but could we also use the limit test for this series?
 

Answers and Replies

  • #2
Dick
Science Advisor
Homework Helper
26,258
618
So we have the series from n=1 to infinite of 1/(n^3+n), and we're supposed to use the integral test, which works out and gives the answer of 1/2 ln2, so it converges.

My question is why cant we use the limit test for this series? Or wait can we? If we take the limit, it would be zero, since we would divide each tearm by n^3 and it would come out to zero.

So basically the book says use the integral test, and thats cool and all, but could we also use the limit test for this series?
What 'limit test' are you talking about? lim a_n->0 doesn't show the sum of a_n converges.
 
  • #3
140
0
Well we're not really looking for the sum, just whether it converges or diverges. We get an actual value using the integral test (which I think is the sum as it goes to infinity?). Using the limit test, a[n]-->0 as n approaches infinity.
 
  • #4
Dick
Science Advisor
Homework Helper
26,258
618
Well we're not really looking for the sum, just whether it converges or diverges. We get an actual value using the integral test (which I think is the sum as it goes to infinity?). Using the limit test, a[n]-->0 as n approaches infinity.
Yes, you ARE looking at the sum. That's the problem you posted. The convergence of the SUM of a sequence (a series) is NOT the same as the convergence of a sequence. Just because both concepts have the word 'convergence' in them doesn't mean they are the same thing. And also, the integral test doesn't give you the sum of the series. It's just a test whether the series converges. The sum may be a different number.
 
  • #5
140
0
Ok so the integral test gives me the sum of the series?

Now if the question were just 'determine whether the series is convergent or divergent', could I have used the limit test?
 
  • #6
Dick
Science Advisor
Homework Helper
26,258
618
Ok so the integral test gives me the sum of the series?

Now if the question were just 'determine whether the series is convergent or divergent', could I have used the limit test?
I've already answered 'no' to both of those. Look, let a_n=1/n. lim a_n converges to 0. sum a_n diverges. They are two different things.
 
  • #7
33,631
5,288
You didn't answer Dick's question asking what the "limit test" is. It might be that you are thinking of what some books call the nth term test for divergence, a test that often confuses students.

Suppose you have a series [itex]\sum a_n[/itex]. The nth term test for divergence says that if
[tex]\lim_{n \to \infty} a_n \neq 0[/tex]
or if this limit doesn't exist,
then the series diverges.

A lot of students misinterpret what this test says, and think mistakenly, that if lim a_n = 0, then the series must converge. This is not what this test is saying. To use the test, the limit of the nth term can't be zero or doesn't exist.
 

Related Threads on Convergence of a Series

Replies
4
Views
2K
  • Last Post
Replies
4
Views
888
  • Last Post
Replies
4
Views
977
  • Last Post
Replies
7
Views
2K
  • Last Post
Replies
24
Views
1K
  • Last Post
Replies
20
Views
1K
  • Last Post
Replies
2
Views
670
  • Last Post
Replies
7
Views
835
  • Last Post
Replies
5
Views
1K
  • Last Post
Replies
2
Views
689
Top