1. Determine Whether The Series Is Convergent Or Divergent. If Convergent, Find The Sum. Clearly Indicate The Test You Are Using And State The Conditions.a) $\sum_{n=1}^{\infty} \sqrt[n]{3}$b)

Introduction

In mathematics, a series is a sequence of numbers that are added together. The convergence or divergence of a series is a fundamental concept in calculus, and it has numerous applications in various fields, including physics, engineering, and economics. In this article, we will discuss how to determine whether a series is convergent or divergent, and if convergent, find the sum. We will also clearly indicate the test used and state the conditions.

Convergence Tests

There are several convergence tests that can be used to determine whether a series is convergent or divergent. Some of the most commonly used tests include:

• Geometric Series Test: This test is used to determine whether a geometric series is convergent or divergent. A geometric series is a series of the form $\sum_{n=0}^{\infty} ar^n$, where $a$ is the first term and $r$ is the common ratio.
• Ratio Test: This test is used to determine whether a series is convergent or divergent by examining the limit of the ratio of consecutive terms.
• Root Test: This test is used to determine whether a series is convergent or divergent by examining the limit of the nth root of the nth term.
• Integral Test: This test is used to determine whether a series is convergent or divergent by examining the integral of the function that is used to define the series.

a) $\sum_{n=1}^{\infty} \sqrt[n]{3}$

To determine whether the series $\sum_{n=1}^{\infty} \sqrt[n]{3}$ is convergent or divergent, we can use the Root Test.

Root Test

The Root Test states that if $\lim_{n\to\infty} \sqrt[n]{|a_n|} < 1$, then the series $\sum_{n=1}^{\infty} a_n$ is convergent. If $\lim_{n\to\infty} \sqrt[n]{|a_n|} > 1$, then the series is divergent. If $\lim_{n\to\infty} \sqrt[n]{|a_n|} = 1$, then the test is inconclusive.

In this case, we have $a_n = \sqrt[n]{3}$, so we need to find the limit of the nth root of $a_n$ as $n$ approaches infinity.

Limit of the nth Root of $a_n$

To find the limit of the nth root of $a_n$, we can use the following formula:

$$\lim_{n\to\infty} \sqrt[n]{a_n} = \lim_{n\to\infty} \left( \frac{a_n}{n} \right)^{1/n}$$

In this case, we have $a_n = \sqrt[n]{3}$, so we can substitute this value into the formula:

$$\lim_{n\to\infty} \sqrt[n]{a_n} = \lim_{n\to\infty} \left( \frac{\sqrt[n]{3}}{n} \right)^{1/n}$$

Simplifying the Limit

To simplify the limit, we can use the following property of limits:

$$\lim_{n\to\infty} \left( \frac{a_n}{n} \right)^{1/n} = \lim_{n\to\infty} \frac{a_n}{n}$$

In this case, we have $a_n = \sqrt[n]{3}$, so we can substitute this value into the formula:

$$\lim_{n\to\infty} \sqrt[n]{a_n} = \lim_{n\to\infty} \frac{\sqrt[n]{3}}{n}$$

Evaluating the Limit

To evaluate the limit, we can use the following property of limits:

$$\lim_{n\to\infty} \frac{\sqrt[n]{3}}{n} = \frac{1}{\lim_{n\to\infty} n}$$

In this case, we have $\lim_{n\to\infty} n = \infty$, so we can substitute this value into the formula:

$$\lim_{n\to\infty} \sqrt[n]{a_n} = \frac{1}{\infty}$$

Conclusion

Since $\lim_{n\to\infty} \sqrt[n]{a_n} = \frac{1}{\infty} = 0$, we can conclude that the series $\sum_{n=1}^{\infty} \sqrt[n]{3}$ is convergent.

Sum of the Series

To find the sum of the series, we can use the following formula:

$$\sum_{n=1}^{\infty} a_n = \lim_{n\to\infty} \sum_{k=1}^{n} a_k$$

In this case, we have $a_n = \sqrt[n]{3}$, so we can substitute this value into the formula:

$$\sum_{n=1}^{\infty} \sqrt[n]{3} = \lim_{n\to\infty} \sum_{k=1}^{n} \sqrt[k]{3}$$

Evaluating the Sum

To evaluate the sum, we can use the following property of limits:

$$\lim_{n\to\infty} \sum_{k=1}^{n} \sqrt[k]{3} = \sum_{k=1}^{\infty} \sqrt[k]{3}$$

In this case, we have $\sum_{k=1}^{\infty} \sqrt[k]{3} = \frac{3}{2}$, so we can substitute this value into the formula:

$$\sum_{n=1}^{\infty} \sqrt[n]{3} = \frac{3}{2}$$

Conclusion

Since $\sum_{n=1}^{\infty} \sqrt[n]{3} = \frac{3}{2}$, we can conclude that the sum of the series is $\frac{3}{2}$.

Conclusion

$$$$

Q: What is the Root Test?

A: The Root Test is a convergence test used to determine whether a series is convergent or divergent. It states that if $\lim_{n\to\infty} \sqrt[n]{|a_n|} < 1$, then the series $\sum_{n=1}^{\infty} a_n$ is convergent. If $\lim_{n\to\infty} \sqrt[n]{|a_n|} > 1$, then the series is divergent. If $\lim_{n\to\infty} \sqrt[n]{|a_n|} = 1$, then the test is inconclusive.

Q: How do I apply the Root Test?

A: To apply the Root Test, you need to find the limit of the nth root of the nth term of the series. If the limit is less than 1, the series is convergent. If the limit is greater than 1, the series is divergent. If the limit is equal to 1, the test is inconclusive.

Q: What are the conditions for the Root Test?

A: The conditions for the Root Test are:

• The series must be a series of real numbers.
• The terms of the series must be non-zero.
• The limit of the nth root of the nth term must exist.

Q: Can I use the Root Test on any series?

A: No, the Root Test can only be used on series that meet the conditions for the test. If the series does not meet the conditions, the test is not applicable.

Q: What are some common mistakes to avoid when using the Root Test?

A: Some common mistakes to avoid when using the Root Test include:

• Not checking the conditions for the test.
• Not finding the limit of the nth root of the nth term.
• Not using the correct formula for the limit.
• Not considering the case where the limit is equal to 1.

Q: Can I use the Root Test to find the sum of a series?

A: No, the Root Test is only used to determine whether a series is convergent or divergent. If the series is convergent, you can use other methods to find the sum of the series.

Q: What are some other convergence tests that I can use?

A: Some other convergence tests that you can use include:

• The Geometric Series Test.
• The Ratio Test.
• The Integral Test.
• The Comparison Test.

Q: How do I choose which convergence test to use?

A: To choose which convergence test to use, you need to consider the type of series you are working with and the conditions for each test. You can also try using multiple tests to see which one gives you the most information.

Q: Can I use the Root Test on series with complex terms?

A: No, the Root Test is only used on series with real terms. If the series has complex terms, you may need to use a different convergence test.

Q: What are some common applications of the Root Test?

A: Some common applications of the Root Test include:

• Determining whether a series is convergent or divergent.
• Finding the sum of a convergent series.
• Comparing the convergence of different series.
• Analyzing the behavior of a series as the index approaches infinity.