The Divergence Test: The Big Idea here is that if the individual terms of an infinite series don't get small, then there's no hope of the series converging! Remember the word "series" means you're adding a bunch of numbers up.
Add up 1 + 1 + 1 + 1 + ... and off you go to infinity!
Add up 1 + 4 + 9 + 16 + ... and off you go to infinity, at an even faster rate!
But add up 1 + 1/2 + 1/4 + 1/8 + ... and as we've seen, this just adds up to be 2.
Notice the terms in the third sequence are "getting small" as we go out further out.
So it's a NECESSARY condition for the terms to get small in order to have the series converge.
As it turns out, though, it's not enough to have the terms get small. The terms have to get small FAST, as opposed to dawdling on their way to zero. Because of this, we say that the terms going to zero is not a SUFFICIENT condition for convergence.
Video: The Divergence Test
The Integral Test: Since the Divergence Test will only tell us about divergence, we need to develop tests that can test conclusively for convergence.
We've already seen that if our series is Geometric, then we can just look at the r = common ratio to see whether the series converges or diverges.
In addition to the Geometric Series Test (from the previous section) we will eventually learn several other tests througout the next sections.
In this section we'll leverage our knowledge of Improper Integrals to develop and use The Integral Test.
Videos: The Integral Test...the reasoning behind why it works
The Integral Test...formal theorem and practice
A worked example (by Krista King)
p-Series: This is one of the easiest tests of all for convergence or divergence of a series!
Video: The Harmonic Series and the p-Series Test
Examples for the p-Series Test
Properties of Infinite Series: Leaving the tests behind for a while, we'll look to see what effect adding, subtracting, or multiplication by a constant has on convergence of series.
***coming up****
Assigned Problems from Textbook
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1st Edition Section 9.4 pg. 659 Divergence Test: 9, 10, 11, 12, 13, Integral Test: 21, 23, 25, p-Series: 29, 31, 32, 33 3rd Edition Section 10.4 pg. 680 Divergence Test: 9, 10, 11, 13, 14, Integral Test: 17, 19, 21, p-Series: 23, 24, 29, 37 |