Welcome back to Unit 10 of AP Calculus BC! Today, we’re going to discuss the nth-term test for divergence with series. Let’s get started!
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This is an AP Calculus BC topic only! If you are taking Calculus AB, you can skip this material. If you’re taking AP Calculus BC, here you go! ⬇️
🤷♀️ What is the nth Term Test for Divergence?
As the name suggests the nth Divergence test tells us if a series will diverge! (mind-blowing stuff guys, I know 🤯). The Divergence test states that:
ifn→∞liman=0,∑andiverges
As we can see, if the nth term doesn't approach 0, the series diverges. On the other hand, if the nth term approaches 0, it creates a situation where the series might converge or still diverge. The crucial point here is that the fate of the series hinges on whether the nth term tends towards zero or not.
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If a series an passes the divergence test, we cannot say the series converges.
🤓 Divergence Test Walkthrough
Let’s try a practice problem together! There are really only 3 steps involved with this:
✏️ Convert to limit notation.
📏 Evaluate the limit.
🤔 Make your conclusion based on the nth-term test.
In conclusion, the for Divergence is a powerful tool for determining whether a series diverges. Remember, if the limit of the nth term does not approach zero, the series diverges. However, passing the divergence test doesn't provide information about convergence. Good luck! 🍀
Key Terms to Review (1)
Nth Term Test: The nth Term Test is a method used to determine whether an infinite series converges or diverges by examining the behavior of its individual terms. If the limit of the sequence of terms as n approaches infinity is not zero, then the series diverges. If the limit is zero, further tests are needed to determine convergence.