Introduction

When evaluating the correctness of mathematical summations, it's crucial to consider all applicable tests and provide a comprehensive analysis. In this article, we will delve into two summations and explore why they were originally deemed incorrect, focusing on the root and ratio tests. By the end of this discussion, you will understand how to correctly identify and validate the summation series.

Understanding the Problems

The two summations in question were initially incomplete because they did not account for all potential valid options. To provide a full solution, we need to use the root and ratio tests to ensure a comprehensive evaluation.

Summation 1: Using the Root Test

Let's first consider the summation series:

1. ( sum_{n1}^{infty} left( frac{1}{2} right)^n )

The root test involves evaluating the limit:

( L lim_{n to infty} left| a_n right|^{1/n} )

For this summation:

( a_n left( frac{1}{2} right)^n )

( left| a_n right|^{1/n} left( left( frac{1}{2} right)^n right)^{1/n} frac{1}{2} )

Thus, ( L frac{1}{2} ). Since ( L

Summation 2: Using the Ratio Test

Now, consider another summation series:

2. ( sum_{n1}^{infty} frac{n!}{3^n} )

The ratio test involves evaluating the limit:

( L lim_{n to infty} left| frac{a_{n 1}}{a_n} right| )

For this summation:

( a_n frac{n!}{3^n} )

( a_{n 1} frac{(n 1)!}{3^{n 1}} )

( frac{a_{n 1}}{a_n} frac{(n 1)!}{3^{n 1}} cdot frac{3^n}{n!} frac{(n 1)}{3} )

( L lim_{n to infty} frac{n 1}{3} infty )

Since ( L > 1 ), the series diverges.

Conclusion: Why the Original Answers Were Missing

The original solution was incomplete because it did not account for the root and ratio tests, which are powerful tools for determining the convergence or divergence of a series. By using these tests, we can ensure a comprehensive evaluation and identify all the correct options.

To avoid such omissions, it's essential to familiarize yourself with multiple methods of analysis, such as the root test and the ratio test, and always consider all possible valid options. This approach will help you ensure the correctness of your answers in similar mathematical problems.

Additional Resources

For further reading and practice, refer to the following resources:

Understanding the Root and Ratio Tests Convergence and Divergence of Series Khan Academy: Convergence and Divergence of Series

By utilizing these resources and understanding the principles behind these tests, you can enhance your problem-solving skills and ensure thorough analysis in your mathematical studies.

Keywords: root test, ratio test, summation correctness