B Difficulty with understanding whether 1/n converges or diverges

[Travis Enigma]

Hi,
I have a quick question about whether or not the infinite series of 1/n converges or diverges. My textbook tells me that it diverges,

but my textbook also says that by the nth term test if we take the limit from n to infinity of a series, if the limit value is not equal to zero the series diverges.

If we take the limit of 1/n, as n approached infinity shouldn't the series actually converge?

Obviously, I know that I'm wrong, but what's wrong with my justification. Does the nth term test not apply here?

Thanks

 

[Infrared]

"If $\sum_{n=1}^\infty a_n$ converges, then $\lim_{n\to\infty}a_n=0$" is correct.

"If $\lim_{n\to\infty}a_n=0$, then $\sum_{n=1}^{\infty}a_n$ converges" is not correct.

If it's snowing, then it must be cold outside. But just because it's cold, it need not be snowing.

 

[Drakkith]

The nth term test does not guarantee convergence, it merely guarantees divergence if the limit of the terms is not zero or does not exist. You must use another test to determine if the series converges.

See here: https://en.wikipedia.org/wiki/Term_test

 

[Travis Enigma]

But the Limit is zero, therefore by the nth term test we know it doesn't diverge. Right?

We may not know whether or not it converges, but it definitely doesn't diverge since the limit is zero and in order for it to diverge it must be nonzero?

 

[Infrared]

But the Limit is zero, therefore by the nth term test we know it doesn't diverge. Right?

No. That's not what the test says. See my post 2.

 

[Drakkith]

But the Limit is zero, therefore by the nth term test we know it doesn't diverge. Right?

No. If the limit of the terms is zero, then the test is inconclusive and you must use another test.

 

[Travis Enigma]

I understand now, thank you so much.

 

[Svein]

For a detailed explanation:

 

[PeroK]

I understand now, thank you so much.

Some tests involve equivalence of two things. For example:

$n$ is an even integer if and only if $n = 2k$ for some integer $k$

That test tells you that both:

1) If $n$ is even, then it can be written as $n = 2k$.

2) If $n$ can be written as $n = 2k$, then $n$ is even.

Some tests, however, only give you the "if" not the "only if". This applies to the divergence test:

If $\lim_{n \rightarrow \infty} a_n \ne 0$, then $\sum_{n= 1}^{\infty} a_n$ diverges. This test does not tell you:

If $\sum_{n= 1}^{\infty} a_n$ diverges, then $\lim_{n \rightarrow \infty} a_n \ne 0$. (WRONG!)

Which is the same as:

If $\lim_{n \rightarrow \infty} a_n = 0$, then $\sum_{n= 1}^{\infty} a_n$ converges. (WRONG!)

Another way to look at this is to identify the case where something is undecided or needs further tests:

1) If $\lim_{n \rightarrow \infty} a_n \ne 0$, then $\sum_{n= 1}^{\infty} a_n$ diverges.

2) If $\lim_{n \rightarrow \infty} a_n = 0$, then we need further tests to decide whether $\sum_{n= 1}^{\infty} a_n$ converges or diverges.

 

[Drakkith]

For a detailed explanation:

I love Mathologer!

 

[Mark44]

But the Limit is zero, therefore by the nth term test we know it doesn't diverge. Right?

At the risk of repeating what others have said, many calculus textbooks refer to this test as the "Nth Term Test for Divergence," meaning that the test can be used to determine whether a series diverges. It cannot be used to determine that a given series converges.

There are two possible results:

1. The series diverges -- this is the conclusion if $\lim_{n \to \infty} a_n \ne 0$ or the limit doesn't exist.
2. The test is inconclusive -- this is the finding if $\lim_{n \to \infty} a_n = 0$. The series could converge (e.g. $\sum \frac 1{n^2}$) or it could diverge (e.g. $\sum \frac 1 n$).

 

[Office_Shredder]

To make a stretched analogy, imagine:
The four sides test: if a polygon does not have four sides, then it's not a square.

If you have a polygon with four sides, this does not tell you if it's a square or not.

The test you are looking at here is similar. It sometimes tells you something does not converge. Other times it tells you nothing.