Summable Sequences: Is {(-1)^n} Summable?

[alovesong]

Homework Statement

Determine whether or not the sequences below are summable:

{(-1)^n}

{(-1)^n + (-1)^(n+1)}

{(-1)^n} + {(-1)^(n+1)}

Homework Equations

The Attempt at a Solution

Okay, I'm having some trouble thinking about these the right way. Since

{(-1)^n}= -1, 1, -1, 1, ... then its sum = -1, 0, -1, 0, -1.

I think this means that it is not summable even though it is 0 every other term.

Assuming it is divergent, then {(-1)^(n+1)} is of course also divergent... But I think that two divergent sequences added together might be convergent.

But does it matter whether they are summed together as one sequence or two? Either way they will still = 0, 0, 0, 0, 0 ... right? So would they both be summable? Sorry if I sound confused - it's just because I am.

 

[tiny-tim]

… you can't add things if they don't exist …

But does it matter whether they are summed together as one sequence or two? Either way they will still = 0, 0, 0, 0, 0 ... right? So would they both be summable? Sorry if I sound confused - it's just because I am.

Hi alovesong!

(it's ok so long as you know you're confused! )

Yes it does matter.

… you can't add things if they don't exist … ​

∑{An} + ∑{Bn} is only defined if both ∑{An} and ∑{Bn} are defined.

Even though ∑{An + Bn} is defined!