Convergence of Series with Variable Exponent and Negative Power

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The second factor goes to zero.In summary, the conversation is about determining the convergence/divergence of a series involving powers and division. The person has tried using the Ratio and Root tests, as well as the Leibniz criterion, but has had no success. Another person suggests using the alternating series test and shows how to prove the limit is 0. The first factor has a finite limit and the second goes to zero, indicating convergence of the series.
  • #1
mikethemike
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Homework Statement



Decide on the convergance/divergance of the following series:

Sum(n=1 to n= infinity) ((n+1)^(n-1))/(-n)^n

where ^ is to the power of and / is divided by.



2. The attempt at a solution

I've used both the Ratio and Root test which are inconclusive (ie. R=1, K=1). Tried changing it around to fit the Leibiniz criterion (and failed). I'm not sure where to go from here...


Thanks for any help :biggrin:

Mike
 
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  • #2
Don't give up on the alternating series test. Take the absolute value. Can you show the limit is 0? Now try to show it's decreasing by showing the derivative of the log is negative.
 
  • #3
Hey Dick,

I'm having trouble convincing myself that the limit is in fact, zero. I can't seem to prove this, even with the absolute value. Can you point me in the right direction?

Thanks

Mike
 
  • #4
Sure. Break it into (n+1)^(n-1)/n^(n-1) times 1/n. The first factor has a finite limit. Can you find it?
 

FAQ: Convergence of Series with Variable Exponent and Negative Power

What is the definition of a convergent series?

A convergent series is a series in mathematics that has a finite sum, meaning that as the number of terms in the series increases, the sum approaches a fixed number. In other words, the terms of the series become smaller and smaller as the series progresses, eventually approaching zero.

What is the definition of a divergent series?

A divergent series is a series in mathematics that has an infinite sum, meaning that as the number of terms in the series increases, the sum increases without bound. In other words, the terms of the series do not approach zero, and the series does not have a finite sum.

What is the difference between a convergent and divergent series?

The main difference between a convergent and divergent series is the sum of the series. A convergent series has a finite sum, while a divergent series has an infinite sum. Another difference is in the behavior of the terms of the series – in a convergent series, the terms become smaller and smaller, while in a divergent series, the terms either do not approach zero or they alternate between positive and negative values.

How can you determine if a series converges or diverges?

There are several tests that can be used to determine if a series converges or diverges, including the comparison test, the ratio test, and the integral test. These tests compare the given series to known series with known convergence or divergence properties to determine the behavior of the given series.

What are some real-world applications of convergent and divergent series?

Convergent and divergent series have many applications in science and engineering, including in the study of electricity and magnetism, fluid dynamics, and signal processing. These concepts are also used in economics and finance, such as in the calculation of compound interest and the valuation of financial assets. Additionally, convergent and divergent series are used in computer science and data analysis to estimate values and make predictions based on large sets of data.

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