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

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In summary, the conversation discusses the summability of several sequences, with a particular focus on the sequences {(-1)^n}, {(-1)^n + (-1)^(n+1)}, and {(-1)^n} + {(-1)^(n+1)}. The confusion arises from the fact that these sequences alternate between values of -1 and 0. While it may seem that they are summable, it is important to note that the definition of summability only applies when both sequences are defined. As such, it is concluded that these sequences are not summable.
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alovesong
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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.
 
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… you can't add things if they don't exist …

alovesong said:
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! :smile:

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

Yes it does matter.

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

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

Even though ∑{An + Bn} is defined! :smile:
 

1. What is a summable sequence?

A summable sequence is a sequence of numbers in which the partial sums converge to a finite limit. This means that as you add more and more terms in the sequence, the sum will eventually approach a fixed value.

2. How do you determine if a sequence is summable?

A sequence is summable if and only if the partial sums converge to a finite limit. This can be determined by calculating the limit of the partial sums and seeing if it approaches a finite value as the number of terms in the sequence increases.

3. What is the sequence {(-1)^n}?

The sequence {(-1)^n} is an alternating sequence where the terms alternate between positive and negative values. It starts with 1 and then alternates between -1, 1, -1, and so on.

4. Is {(-1)^n} summable?

Yes, {(-1)^n} is summable. The partial sums of this sequence alternate between 0 and 1, but they eventually converge to 0 as the number of terms increases. This means that the sequence is summable and the sum is 0.

5. How is the summability of {(-1)^n} related to the alternating series test?

The alternating series test states that if a series alternates between positive and negative terms and the absolute values of the terms decrease as n increases, then the series is convergent. Since {(-1)^n} satisfies these conditions, it is summable by the alternating series test.

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