Radius and Interval of Convergence for (3^n x^n)/(n+1)^2 Series

In summary, the conversation discusses finding the radius and interval of convergence for a series, specifically (3^n x^n) / (n+1)^2. The ratio test was used to determine the radius of convergence to be 1/3. To find the interval of convergence, the values -1/3 and 1/3 were plugged into x and it was determined that the series converges at 1/3 due to the p-series test. However, for -1/3, the alternating series test or relating it to the series at 1/3 could be used to show convergence. The conversation also mentions that the book does not mention using absolute values in the alternating series test.
  • #1
badtwistoffate
81
0
Have to find the radius of convergence and interval of convergence,
the series is (3^n x^n ) / (n+1)^2,
did the ratio test and found the radius of convergence to be the 1/3.
now for finding the interval of convergence I plug in -1/3 and 1/3 into x and find out if it converges or not

For 1/3, it converges due to p-series, 2>1.

But for -1/3 I know it converges but can see why? Any help here at this endpoint?
 
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  • #2
You could use the alternating series test.

More simply, you can relate the series at -1/3 to the series at 1/3...
 
  • #3
I don't quite understand, the alternating series only works when An+1< or = to An, and in that series it doesnt, because An is negative and An+1 is postivie?
Could you elaborate and how I would relate it to the series at 1/3?
 
  • #4
Look at the alternating series test again, it's the absolute values of the terms that are decreasing (and going to zero) while the sign is alternating.


The series at 1/3 is the absolute values of the terms of the series at -1/3, i.e. you've already should that the series at -1/3 is absolutely convergent.
 
  • #5
iiiiiiiiiii...
my book doesn't say absolute value... so idk. I see what you mean if that's true. Why doesn't my book say that it says just that its decreasing or equal too...
 

What is the "Quickie Ratio Test Question"?

The "Quickie Ratio Test Question" is a mathematical concept used in calculus to determine the convergence or divergence of an infinite series. It involves taking the ratio of consecutive terms in a series and using that information to determine if the series converges or diverges.

How do you perform the "Quickie Ratio Test"?

To perform the "Quickie Ratio Test," you take the absolute value of the ratio of consecutive terms in a series. If this ratio approaches a finite number as the terms in the series approach infinity, then the series converges. If the ratio approaches infinity or does not approach a finite number, then the series diverges.

What is the purpose of the "Quickie Ratio Test"?

The purpose of the "Quickie Ratio Test" is to determine the convergence or divergence of an infinite series. This information is important in many areas of science and mathematics, as it allows us to make predictions and draw conclusions about the behavior of a series.

When is the "Quickie Ratio Test" most commonly used?

The "Quickie Ratio Test" is most commonly used when the terms of a series involve factorials, exponential functions, or other complicated expressions. It is also often used when the terms of a series involve powers of n or other variables.

Can the "Quickie Ratio Test" be used on all infinite series?

No, the "Quickie Ratio Test" cannot be used on all infinite series. It is only applicable to series with positive terms, and there are other tests that should be used for alternating series or series with negative terms. It is important to carefully consider the terms of a series before applying the "Quickie Ratio Test."

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