Alternating Series Test Example SUM( (-1)^(n + 1)/(n + 7)) | Summary and Q&A

TL;DR

This video explains the process of determining if an alternating series converges or diverges using the alternating series test.

Key Insights

• 🏆 The alternating series test is used to determine the convergence or divergence of an alternating series based on specific conditions.
• 🏆 The first step of the test involves identifying the non-alternating part of the series, denoted as "a sub n."
• 🏆 The second step of the test requires checking if the limit of the non-alternating part as n approaches infinity is zero.
• 🍉 Additionally, it is necessary to verify if the series is non-increasing, meaning the terms are decreasing or staying the same as n increases.
• 🏆 The alternating series test can only determine convergence and not divergence.
• 🏆 If any of the conditions of the alternating series test are not satisfied, other tests, such as the nth term test, should be used.
• ❓ It is not always necessary to take the derivative to show that a series is decreasing; it can often be observed visually.

Transcript

in this problem we have an infinite sum and we're being asked to determine if it converges or diverges so this is an alternating series because it has a negative one to the n plus one so we can use what's called the alternating series test so to use the alternating series test the first thing you have to do is identify your a sub n so in the altern... Read More

Questions & Answers

Q: What test is used to determine if an alternating series converges or diverges?

The alternating series test is used to determine if an alternating series converges or diverges. It involves checking the behavior of the non-alternating part of the series and verifying if certain conditions are met.

Q: How do you identify the non-alternating part of an alternating series?

In the alternating series test, the non-alternating part is referred to as "a sub n." It can be identified by separating the alternating and non-alternating parts of the series and observing which part remains constant.

Q: What does it mean for a series to be non-increasing?

When a series is non-increasing, it means that the terms of the series are either decreasing or remaining the same as n increases. This condition is important in the alternating series test to determine convergence.

Q: What should be done if one of the conditions in the alternating series test is not satisfied?

If one of the conditions in the alternating series test is not satisfied, the test fails to determine convergence or divergence. In such cases, it is necessary to revert to other tests, such as the nth term test, to evaluate the series.

Summary & Key Takeaways

• The video discusses the application of the alternating series test to determine the convergence or divergence of an infinite sum.

• The first step in the alternating series test is identifying the non-alternating part of the series.

• To use the test, one must check if the non-alternating part approaches zero as n goes to infinity and if the series is non-increasing.