# I was wondering why this works and if it works every time

• end3r7
In summary, the conversation discusses using the ratio and root tests to determine the values of x for which a series will converge, and the use of these tests for power series. The conversation also confirms that the use of these tests is not limited to power functions.
end3r7

## Homework Statement

I have to find for which values of x the following converge

## Homework Equations

$$\sum n x^{n}$$

$$\sum \frac{x^{n}}{n}$$

$$\sum n^{n} x^{x}$$

## The Attempt at a Solution

I used the ratio test for the first two and the root test for the last and found respectively that x must lie within

(-1,1)
[-1,1)
{0}

Are these right?
and
Is this the right way to go all the time?

I was wondering when I'll be able to apply the ratio/root tests for radii of convergence. Is it just when we have power functions?

For the last one you mean sum(n^n*x^n), right? I think they are all correct. You can apply any legitimate test to any series, whether it is a power function or not.

The ratio and root test work for any series, as Dick said, but it may be difficult to calculate the ratios or roots. It is when you have products or powers, as in power series, that they are easy to apply.

You are correct Dick.

And thanks for the advice guys. =D

## 1. Why does this work?

This is a common question in science and it is important to understand the underlying mechanisms and principles behind any scientific phenomenon. In order to answer this question, scientists conduct experiments and gather data to analyze and make conclusions about why something works in a certain way.

## 2. How does it work?

Similar to the first question, this question also seeks to understand the mechanisms and processes involved in a scientific phenomenon. Scientists use various techniques and methods such as observation, experimentation, and modeling to determine how something works.

## 3. Is this a reliable result?

This question is important in ensuring the validity and credibility of any scientific finding. Scientists use statistical analysis and rigorous methods to ensure that their results are reliable and can be replicated by others in the scientific community.

## 4. Can this be applied in other situations?

Many scientific discoveries and theories have practical applications that can benefit society. This question is often asked to determine the potential uses and implications of a scientific finding. Scientists also conduct further research to explore potential applications of their findings.

## 5. Does this work every time?

This is a crucial question in understanding the limitations and scope of a scientific concept or theory. While many scientific principles are generally accepted to work most of the time, there may be exceptions or specific conditions where they do not apply. Scientists continue to conduct research to understand the factors that may affect the reliability of their findings.

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