Can alternating series be grouped as geometric series for convergence?

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In summary, the conversation discusses reviewing power series for use in differential equations and dealing with alternating series. The example of \sum(-1)^n(\frac{3}{2})^n is given, which fails the alternate series test due to the limit of a_n not equaling 0. The question is raised about grouping (-1)^n into the fraction and considering it as a geometric series, but it is concluded that the series still diverges. It is then stated that if the summand doesn't go to zero, the series cannot converge. The conversation ends with agreement that this is the best approach to solving the problem.
  • #1
kdinser
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I'm reviewing power series for use in differential equations and I'm having some trouble remembering how to deal with alternating series.

For instance, if I have:
[tex]\sum(-1)^n(\frac{3}{2})^n[/tex]

if [tex]a_n=(\frac{3}{2})^n[/tex]
This fails the alternate series test because the limit of [tex]a_n[/tex] as n goes to infinity doesn't equal 0.

Can I group the (-1)^n into the fraction and call it a geometric series? In that case, it would diverge, |r| would be greater then 1.
 
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  • #2
kdinser said:
I'm reviewing power series for use in differential equations and I'm having some trouble remembering how to deal with alternating series.

For instance, if I have:
[tex]\sum(-1)^n(\frac{3}{2})^n[/tex]

if [tex]a_n=(\frac{3}{2})^n[/tex]
This fails the alternate series test because the limit of [tex]a_n[/tex] as n goes to infinity doesn't equal 0.

Can I group the (-1)^n into the fraction and call it a geometric series? In that case, it would diverge, |r| would be greater then 1.

Yes, both ways are fine.
 
  • #3
thanks for the help.
 
  • #4
If the summand doesn't go to zero, the series cannot converge, regardless of whether it is alternating or not (using the most common definition of convergence).
 
  • #5
Data said:
If the summand doesn't go to zero, the series cannot converge, regardless of whether it is alternating or not (using the most common definition of convergence).

Yes, you're right. This is the best way to solve the problem. The summand doesn't go to zero so the series diverges.
 

1. What is an alternate series?

An alternate series is a mathematical series in which the terms alternate between positive and negative values. This means that every other term in the series is positive, while the remaining terms are negative.

2. How is an alternate series different from a regular series?

An alternate series differs from a regular series in that the terms alternate between positive and negative values, while a regular series has all positive or all negative terms.

3. What is the alternating series test?

The alternating series test is a method used to determine whether an alternate series converges or diverges. It states that if the terms of an alternate series decrease in absolute value and approach 0, then the series will converge.

4. How can the alternating series test be applied to determine convergence?

To apply the alternating series test, you must first check if the terms of the alternate series decrease in absolute value and approach 0. If they do, then the series will converge. If not, then the series will diverge.

5. Are there any exceptions to the alternating series test?

Yes, there are a few exceptions to the alternating series test. One exception is when the terms do not decrease in absolute value and approach 0, but the series still converges. Another exception is when the terms decrease in absolute value and approach 0, but the series still diverges. In these cases, other tests or methods must be used to determine convergence or divergence.

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