Divergence Test For Series - Calculus 2

TL;DR

The divergence test allows you to determine if a series will diverge or not based on the limit of a sequence. If the limit is not zero, the series diverges.

Transcript

in this video we're going to talk about the divergence test so what's the main idea behind this test the purpose of this test is to tell you if a series will diverge so here's what you need to know if you take the limit as n approaches infinity of some sequence a sub n if it does not equal 0 then the series the series is going to diverge now what a... Read More

Key Insights

• 🏆 The divergence test determines if a series will diverge based on the limit of a sequence.
• ⛔ If the limit of the sequence is not equal to zero, the series diverges.
• 🏆 The divergence test is a simple and efficient method to determine divergence.
• 🥳 Geometric series with a common ratio greater than one will always diverge.
• ❓ The harmonic series is an example of a series that always diverges.
• 🥳 Geometric series with a common ratio less than one will converge.
• 🍹 The sum of an infinite geometric series can be calculated using the formula: sum = a / (1 - r).

Questions & Answers

Q: What is the purpose of the divergence test?

The divergence test is used to determine if a series will diverge or not based on the limit of a sequence. If the limit is not zero, the series diverges.

Q: What happens if the limit of the sequence is equal to zero?

If the limit of the sequence is zero, the series may converge or it may also diverge. Further tests are required to determine its convergence or divergence.

Q: How can you apply the divergence test to a specific series?

To apply the divergence test, calculate the limit of the sequence by taking the limit as n approaches infinity. If the limit is not equal to zero, the series automatically diverges.

Q: What is the result of the divergence test for the series n/(3n-4)?

By applying the divergence test, it is determined that the series n/(3n-4) diverges. This is because the limit of the sequence is 2/3, which is not equal to zero.

Summary & Key Takeaways

• The divergence test is used to determine if a series will diverge or not based on the limit of a sequence.

• If the limit of the sequence is not equal to zero, the series automatically diverges.

• Geometric series with a common ratio less than one will converge, while those with a ratio greater than one will diverge.