Can we use the limit test for this series?

[mohabitar]

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 can't 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 that's cool and all, but could we also use the limit test for this series?

 

[Dick]

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 can't 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 that's 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.

 

[mohabitar]

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.

 

[Dick]

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.

 

[mohabitar]

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?

 

[Dick]

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.

 

[Mark44]

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 $\sum a_n$. The nth term test for divergence says that if
$$\lim_{n \to \infty} a_n \neq 0$$
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.