10.6.2 converge or diverge? alternating series

In summary, convergence and divergence refer to the behavior of an infinite series, where the terms either approach a specific value or do not have a definite limit. If an alternating series has non-increasing terms, the alternating series test can be used to determine its convergence or divergence. However, an alternating series with positive terms cannot converge. Other tests, such as the ratio test and the root test, can also be used to determine convergence or divergence. An alternating series can also diverge to infinity if the absolute value of the terms does not approach zero and the series does not satisfy the conditions of Leibniz's test.
  • #1
karush
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MHB
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converge or diverge
$$S_n= \sum_{n=1}^{\infty} (-1)^{n+1}\frac{\sqrt{n}+6}{n+4}$$
ok by graph the first 10 terms it looks alterations are converging to 0
 
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  • #2
alternating series whose nth term $\rightarrow 0 \implies$ convergence
 

FAQ: 10.6.2 converge or diverge? alternating series

What does it mean for a series to converge or diverge?

When talking about a series, convergence refers to the idea that the terms of the series eventually get closer and closer to a specific value as more terms are added. Divergence, on the other hand, means that the terms of the series do not approach a specific value and the series does not have a finite sum.

How do you determine if an alternating series converges or diverges?

To determine if an alternating series converges or diverges, you can use the Alternating Series Test. This test states that if the absolute value of the terms in the series decrease as more terms are added and eventually approach 0, then the series will converge.

What is the difference between absolute and conditional convergence?

Absolute convergence refers to a series where the absolute value of the terms converges. This means that the series will also converge. On the other hand, conditional convergence refers to a series where the absolute value of the terms diverges, but the series itself still converges. This is only possible for alternating series.

Can an alternating series converge to a negative value?

Yes, an alternating series can converge to a negative value. This is because the alternating series test only requires that the absolute value of the terms approach 0, not the actual value of the terms. So, as long as the terms alternate between positive and negative values, the series can converge to a negative value.

What are some real-world applications of alternating series?

Alternating series have many real-world applications, including in physics, engineering, and economics. For example, alternating series can be used to model the oscillations of a pendulum, the voltage of an alternating current circuit, or the fluctuations in stock prices. They can also be used in numerical methods to approximate solutions to equations that do not have exact solutions.

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