Determining Convergence of Alternative Series

In summary, convergence of an alternative series is determined by the behavior of the series as the number of terms increases. This can be determined by applying one of several convergence tests such as the alternating series test, the ratio test, or the root test. The alternating series test states that if the terms of an alternative series decrease in absolute value and approach zero, then the series is convergent. The ratio test compares the ratio of consecutive terms to a limit, while the root test compares the nth root of the absolute value of the nth term to a limit. Both the ratio and root tests can determine if a series is convergent or divergent, but may be inconclusive in certain cases.
  • #1
quasar987
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Leibniz criterion for alterning serie say that if the two conditons a_n >0 is decreasing and -->0 are satisfied, the serie converges. It doesn't say that if they don't it diverge.

So how do you determine the convergence of an alternative serie that doesn't satisfy the conditions? For exemple,

[tex]\sum_{n=1}^{\infty} (-1)^n\frac{1}{n^{1/n}}[/tex]

[tex]a_n \rightarrow 1 \neq 0[/tex]
 
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  • #2
So, you're asking what test to use when the terms don't converge to 0?
 
  • #3
Yeah, a test, ok.. maybe I should have tought about this one a little longer. :/
 

What is the definition of convergence for alternative series?

Convergence of an alternative series is determined by the behavior of the series as the number of terms increases. If the series approaches a finite limit, it is said to be convergent. If it does not approach a finite limit, it is said to be divergent.

How is the convergence of an alternative series determined?

The convergence of an alternative series is determined by applying one of several convergence tests, such as the alternating series test, the ratio test, or the root test. These tests examine the behavior of the series to determine if it approaches a finite limit or if it diverges.

What is the alternating series test?

The alternating series test is a method for determining the convergence of an alternative series. It states that if the terms of an alternative series decrease in absolute value and approach zero, then the series is convergent. This test can be used to determine the convergence of many alternating series, but it does not work for all series.

How does the ratio test determine convergence of alternative series?

The ratio test is a convergence test that compares the ratio of consecutive terms in a series to a limit. If the limit is less than 1, the series is convergent. If the limit is greater than 1, the series is divergent. If the limit is equal to 1, the test is inconclusive and another test must be used.

What is the root test and how does it determine convergence of alternative series?

The root test is another convergence test that compares the nth root of the absolute value of the nth term in a series to a limit. If the limit is less than 1, the series is convergent. If the limit is greater than 1, the series is divergent. If the limit is equal to 1, the test is inconclusive and another test must be used. This test is often used for series with factorials or exponential terms.

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