Convergance and Divergence Could someone go over a routine to help determine them?

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In summary, there are various tests that can be used to determine the convergence or divergence of a series, such as the comparison test, integral test, ratio test, and nth root test. For the specific example given, the comparison test with the series \sum_{n=1}^\infty \frac{1}{n^2} is recommended. Additional resources for learning about these tests can be found at the provided links.
  • #1
tamintl
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Hey there..

Basically I'm struggling with convergence and divergence of series.

I can see if converges and diverges by common sense and thinking through in my head but I struggle to write it down. The definitions in books seem confusing.

Are there any steps I can systematically do every time which will help me determine on paper if the said series div. or conv.?

For example a basic typical question in my course is like the following:

as n ---> infinity

[itex]\sum[/itex] (n^2 - 1) / (n^4 + 1)

Some detailed exlanation in basic language would be greatly appreciated


Regards
Tam
 
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  • #2
There are many different tools to test the convergence of a series, the most common probably being the comparison test, the integral test, the ratio test, and the nth root test. Here's a link with a big list: http://www.math.hmc.edu/calculus/tutorials/convergence/

In your particular example, I think comparison with the series [itex] \sum_{n=1}^\infty \frac{1}{n^2}[/itex] is the easiest test to use.
 

What is convergance and divergence?

Convergance and divergence are concepts used to describe the behavior of a sequence or series of numbers. Convergance refers to the idea that a sequence or series will approach a specific value as it continues, while divergence means that the sequence or series will not approach a specific value and may instead increase or decrease without bound.

What is the difference between convergance and divergence?

The main difference between convergance and divergence is the behavior of the sequence or series. Convergance indicates that the sequence or series will approach a specific value, while divergence means that the sequence or series will not approach a specific value and may instead increase or decrease without bound.

How can I determine if a sequence or series is convergent or divergent?

There are a few methods that can help determine if a sequence or series is convergent or divergent. One common method is to use the limit test, where you take the limit of the sequence or series and see if it approaches a specific value or not. Another method is to use the ratio test, which looks at the ratio between consecutive terms in the sequence or series. If the ratio is less than 1, the series is convergent, and if it is greater than 1, the series is divergent.

What are some real-life applications of convergance and divergence?

Convergance and divergence can be seen in various real-life applications, such as in finance, physics, and population growth. In finance, convergance and divergence can be used to analyze the behavior of investments and predict future values. In physics, they can be used to model the behavior of systems, such as the movement of planets. And in population growth, they can be used to predict the growth or decline of a population over time.

Is there a specific method or formula to find convergance and divergence?

There is no one specific method or formula to find convergance and divergence. Each sequence or series may require a different approach, and there are multiple tests and techniques that can be used. It is important to understand the different methods and choose the most appropriate one for the specific sequence or series being analyzed.

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