Show ( (-1)^n + (1/n) ) diverges.

[Unassuming]

Homework Statement

Show that the sequence (-1)^n + (1/n) diverges.

Homework Equations

The Attempt at a Solution

I have assumed it converges and found the difference between consecutive terms to be "1+ (1/ 2n(n+1) ). I have tried many things but I cannot progress.

 

[arildno]

Well, your expression for the difference is wrong.
Think otherwise:
Your sequence is a sum of two other sequences.
What can you say about the convergens/divergence of those two subsequences?

 

[Unassuming]

They both converge to two different numbers.

 

[Unassuming]

We have not gotten to the subsequences yet so I hesitate to say that this violates the thrm saying that subsequences always converge to the same point as the sequence itself.

This problem is two sections before that theorem and any talk of subsequences. Is there any way to prove it without them?

 

[HallsofIvy]

They both converge to two different numbers.

And what are the two numbers you think they converge to?

 

[Unassuming]

1 and -1? I am beginning to doubt myself.

 

[danago]

1 and -1? I am beginning to doubt myself.

Try a few different values of n, and see if that helps visualize what is happening to each term. For the (-1)^n part, can i suggest that you see what happens when n=1,2,3,4...Does it really converge to 1?

EDIT: Sorry misunderstood what you were saying, thought that you meant (-1)^n converges to 1 for all even and odd n.

 

[Unassuming]

I know what it does. I have tried values of n. I have pictured it and I am ready to prove it. Am I missing something here?

For all the even n, it converges to 1 from the right.

For all the odd n, it converges to -1 from the right.

I am trying to prove that it diverges.

 

[Gib Z]

Ok. First - separate them into two separate sequences. It in fact, does NOT converge. You took some partial sums, did the partial sums ever tend to a certain value? The partial sums "oscillates" between 1 and -1 . It doesn't converge. For the purpose of the question, we need only know that is it bounded. I'm sure you can do the rest.

 

[HallsofIvy]

1 and -1? I am beginning to doubt myself.

No, neither of them converges to 1 or -1! I have no idea how you could come to that conclusion.

Look at a few terms: {1/n} starts 1, 1/2, 1/3, 1/4, 1/5, ... what does that converge to?

{(-1)ⁿ starts -1, 1, -1, 1, -1, 1... what does that converge to? Does it converge at all?

 

[Gib Z]

O wait sorry, I thought we were doing series instead of sequences. Ignore my comment about the boundedness.