Can the Alternating Series Test Determine Divergence?

In summary, the conversation discusses finding a sequence a_{n} that is both non-negative and null, but where the series \sum (-1)^{n+1} a_{n} is divergent. The alternating series test is mentioned as a possible solution, but the participants are confused about how it could work. A suggestion is made to use a sequence with alternating zeros and positive terms to satisfy the test.
  • #1
kidsmoker
88
0

Homework Statement



Find a sequence [tex]a_{n}[/tex] which is non-negative and null but where [tex]\sum (-1)^{n+1} a_{n}[/tex] is divergent.

Homework Equations



Alternating series test:

Let [tex]a_{n}[/tex] be a decreasing sequence of positive real numbers such that [tex]a_{n}\rightarrowa[/tex] as [tex]n\rightarrow\infty[/tex]. Then the series [tex]\sum (-1)^{n+1} a_{n}[/tex] converges.

The Attempt at a Solution



I'm a bit confused by this one. If [tex]a_{n}[/tex] is non-negative and null then it seems like it's decreasing to zero, in which case it satisfies the alternating series test. So how can the sum diverge?!
 
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  • #2
How about (1,0,1/2,0,1/3,0,1/4,0...)? It's null but nondecreasing. The (-1)^(n+1) doesn't help much does it?
 
  • #3
Ah yeah i see. So you sort of pad it out with zeros to remove the minus terms. Thanks!
 

What is the Alternating Series Test?

The Alternating Series Test is a method used to determine the convergence or divergence of an infinite series that alternates in sign.

How is the Alternating Series Test used?

The Alternating Series Test states that if the terms of an alternating series decrease in absolute value and approach 0, then the series is convergent. This means that the series will approach a finite number as more terms are added.

What is the formula for the Alternating Series Test?

The formula for the Alternating Series Test is: ∑ (-1)^n * a_n, where a_n is the nth term of the series.

What is an example of using the Alternating Series Test?

An example of using the Alternating Series Test is determining the convergence of the series: ∑ (-1)^n/n. By applying the test, we can see that the terms decrease in absolute value and approach 0 as n increases, so the series is convergent.

What is the difference between conditional and absolute convergence in the Alternating Series Test?

Conditional convergence refers to a series that converges, but only under certain conditions, such as the Alternating Series Test. Absolute convergence, on the other hand, refers to a series that converges regardless of the order in which the terms are added. The Alternating Series Test can only determine conditional convergence, not absolute convergence.

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