Finding the convergence of a parametric series

In summary, a parametric series is a mathematical series defined by a pair of parameters or variables. To determine its convergence, the ratio and root test can be used. Finding the convergence of a parametric series is significant for understanding its finite sum and its applications. A parametric series can converge to a negative or complex number if it meets the conditions for convergence. Other methods for finding convergence include the integral, comparison, and alternating series tests, each with its own set of conditions and limitations.
  • #1
Fochina
4
1
Homework Statement
find for what ## \alpha ## the series converges
Relevant Equations
$$\sum_{n}\left ( \sqrt[n]{n}-\sqrt[n]{2} \right )^\alpha $$
It is clear that the terms of the sequence tend to zero when n tends to infinity (for some α) but I cannot find a method that allows me to understand for which of them the sum converges. Neither the root criterion nor that of the relationship seem to work. I tried to replace ##\sqrt[n]{n}## with ##e^{\frac{ln n}{n}}## but then I don't understand how to proceed.
 
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  • #2
The result seems to be ##\alpha \geq 1##. So you could work with a convergent upper and a divergent lower bound which gives you some flexibility to change the function.
 

FAQ: Finding the convergence of a parametric series

What is a parametric series?

A parametric series is a type of mathematical series where the terms are defined by a set of parameters, rather than a fixed pattern. In other words, the terms of the series are dependent on one or more variables.

How do you find the convergence of a parametric series?

To find the convergence of a parametric series, you must first determine the values of the parameters that make the series converge. This can be done by using various convergence tests, such as the ratio test or the root test. If the series converges for a particular set of parameter values, it is said to be convergent for those values.

What is the significance of finding the convergence of a parametric series?

Finding the convergence of a parametric series is important because it allows us to determine the behavior of the series for different values of the parameters. This can help us understand the overall behavior of the series and make predictions about its behavior in different scenarios.

Can a parametric series diverge?

Yes, a parametric series can diverge if the parameters are chosen in such a way that the series does not converge. In other words, there may be certain values of the parameters for which the series does not have a finite limit and therefore diverges.

Are there any real-world applications of finding the convergence of a parametric series?

Yes, there are many real-world applications of finding the convergence of a parametric series. For example, in physics and engineering, parametric series are often used to model real-world phenomena and finding their convergence can help in making predictions and solving problems. Additionally, in economics and finance, parametric series are used to analyze and predict trends in various markets.

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