Exploring Ratio Test Examples | Understanding Notations and Cancellations

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In summary, the ratio test is a mathematical test used to determine the convergence or divergence of a series by taking the limit of the ratio of the terms in the series. It is commonly applied in fields such as physics, engineering, and economics to analyze systems. Understanding notations in the ratio test is crucial for correctly applying the test and interpreting its results, and cancellations in the expressions can make it easier to determine the limit. However, the ratio test has limitations and may not be applicable in all cases, as it can only determine absolute convergence for series with positive terms.
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Homework Statement



I'm attached a pic of an example.

just wondering when i was trying the ratio test with a problem, i noticed numbers displayed in the pic.

is another way to write those examples like this: 5*5 or (x-1)*(x-1) ?

is that all the n+1 means?

also if there was something like: (n^7) / (n+1)^7 how can those cancel?
sorry about notation there.

thanks

Homework Equations





The Attempt at a Solution

 
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  • #2
nevermind.
 
  • #3
[tex] \frac{n^7}{(n+1)^7} [/tex] behaves like [tex] \frac{n^7}{n^7} [/tex] for large n

so in your example it's just 1 if the limit goes to infinity. Can you see it?
 

FAQ: Exploring Ratio Test Examples | Understanding Notations and Cancellations

What is the ratio test?

The ratio test is a mathematical test used to determine the convergence or divergence of a series. It involves taking the limit of the ratio of the terms in the series to determine if the series is convergent or divergent.

How is the ratio test applied in real-world scenarios?

The ratio test is commonly used in fields such as physics, engineering, and economics to analyze the behavior of various systems. For example, it can be used to determine the stability of a chemical reaction or the growth rate of a population.

What is the purpose of understanding notations in the ratio test?

Understanding notations in the ratio test is crucial in correctly applying the test and interpreting its results. Notations such as n, k, and an represent the terms in the series, and correctly identifying them is essential for accurate calculations.

Why are cancellations important in the ratio test?

Cancellations occur when terms in the ratio test cancel each other out, resulting in a simpler expression. These cancellations can make it easier to determine the limit of the ratio and ultimately determine the convergence or divergence of the series.

Are there any limitations to the ratio test?

Yes, the ratio test is not always applicable and may not be able to determine the convergence or divergence of a series. It is best used for series with positive terms and can only determine absolute convergence, not conditional convergence.

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