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Techniques that let you tell whether a series converges are unimaginatively called tests for convergence or convergence tests. Fortunately for us, we can equate them to Pandora's box. Pandora's box was jam-packed with all of the evils of the world and a little surprise. When she opened it, they all escaped, but the little surprise, hope, remained.
For series, we just have numbers and sandwiches in our box. If the box if full of infinite amounts of food, the series diverges. Grilled cheese spills out all over when we open the box. If it has finite amounts, the series converges. The grilled cheese stays put when we open the box.
In this section we'll introduce a lot of convergence tests and show how to use them. We'll also have a great long discussion about how you know when to use which test(s). With our new found knowledge, we'll know better than to open up a divergent Pandora's box of grilled cheese.
Since an uncontrolled grilled cheese spill is a hazardous materials catastrophe we'd like to avoid, we want a test that will tell us when not to open the Pandora's box.
We know that if a series converges, its terms must approach zero. Rephrasing this in the contrapositive: if the terms of a series don't approach zero, the series diverges. This statement lets us look at a series and, if the terms don't approach zero, conclude that the series diverges.
Be Careful: We can't use this statement to conclude that a series converges. We can only use it to evaluate if a series diverges. That's why we call it the Divergence Test. If the terms do approach zero, there's hope that the series might converge, but we would need to use other tools to really draw that conclusion.
Does the series
converge or diverge?
Look at the limit of the terms an = n as n goes to ∞. Since
the series diverges.
The first tool in our arsenal of convergence tests is for alternating series, which is a series whose terms alternate in sign. This is for a Pandora's box full of American and Swiss grilled cheese sandwiches. The alternating series test can tell us if it's safe to open that box.
1 – 2 + 3 – 4 + 5 – 6 + ...
is an alternating series because the signs of the terms switch back and forth between positive and negative.
We can find a formula for the terms of an alternating series the same way we can find a formula for the terms of an alternating sequence.
Line jumping is the idea behind our first convergence test, the alternating series test. Since the terms of an alternating series change sign, the partial sums for any alternating series will jump back and forth over some line. If the terms are getting smaller and approaching zero, the partial sums will get closer to the line and so the series will converge.
Alternating Series Test (AST): If Σ an is an alternating series, and if
|an| > |an + 1|
for all n (that is, the terms have strictly decreasing magnitude), and if
then the series converges.
The catch: We can't use the AST to conclude a series diverges. We can only use the AST to conclude that a series converges. If the AST doesn't tell us that the series converges, we need to use another test, which might include the divergence test.
Can we use the AST to conclude that the series
We have an alternating series. For all n, we have
so the first condition is met. We also have
so the second condition is met. Since all conditions are met, the AST says that the series converges.
Suppose we know, now, that we have an alternating series Σ an that converges, and it converges to value L. Here, L stands for limburger, which is the type of stinky cheese in our Pandora's box. We know we can open it without fear of a sandwich explosion, but it will smell terrible.
A partial sum Sn is just an approximation of L. We'd like to know how good the approximation is—that is, how close are Sn and L? In symbols, we'd like to say something about the value.
|L – Sn|.
This distance between the approximation Sn and the real sum of the series L is called the error. With alternating series, it's awesomely easy to find an upper bound for the error.
In general, suppose we have a convergent alternating limburger series Σ ai that sums to L. Then the error between Sn and the actual sum L can't be more than the absolute value of the (n + 1)st term of the series.
We use the absolute value of the (n + 1)st term because the only thing the sign is good for is saying which way the jump goes.
To find the (n + 1)st term of the series we need to know the starting index of summation, or we won't know which term is the (n + 1)st.
The next tool in our convergence test arsenal is the ratio test. We get the idea from the convergence of geometric series. We mentioned before that geometric series are as common as eating hotdogs. In the case of the ratio test, we want to know if we can safely open our box full of grilled chili-cheese hotdogs.
If Σ an
is a geometric series, we know that the common ratio of the series is
We also know that, if |r| < 1, the geometric series converges.
Now suppose Σ an isn't geometric. Since this series isn't geometric the ratio between successive terms isn't constant, but we're not going to let that stop us. We're going to pretend the series is geometric and look at the ratio between terms anyway. It's like pretending a grilled chili-cheese hotdog is just a hotdog. Pass the wetnaps.
More precisely, we're going to look at the limit of the ratios as n goes to ∞. The ratio test says that, if the ratio of terms
converges to something with absolute value less than 1, the series converges.
Put another way, if the grilled chili-cheese dog isn't too messy, we can eat it without it getting everywhere. It's worth opening the Pandora's box and having a bite.
To show this, we'll stick the absolute value signs inside the limit and look at
In symbols, start with a series
The ratio test says:
All we're doing, really, is pretending that we're looking at a geometric series.
Before practicing the ratio test, you should be very comfortable with the following ideas.
When is the ratio test actually useful? Anytime we encounter a mess of chili and cheese for sure, which corresponds to nasty fractions in terms of math. In particular, we used it for series with fractions where both the numerator and the denominator contained the variable. The ratio test didn't work on the series
where the terms only had the variable in the denominator. The ratio test also didn't work on the series
While there is an n in the numerator, all it does is flip-flop the sign of the term, so this n doesn't really count.
Observation: The ratio test works well on interesting fractions, since we're able to cancel out pieces of the terms to get simpler expressions. These are the ones that easily run out of control, like Velveeta cheese.
Here are the formulas for the terms of the different series on which we were able to use the ratio test:
There are a lot of factorials and exponents in these terms.
Observation: The ratio test works well when the fractions contain factorials and exponents, because we can cancel these out to get simpler expressions. These are the fractions that have so much chili and cheese on them that we need a bowl to catch our drippings as we eat.
We already know that series and integrals share some similar properties. We're going to show that, by replacing the series with an equivalent integral, we can determine if the series converges.
In terms of our Pandora's box, we're just replacing our grilled cheese with something else. Assume we've replaced them with gremlins. Maybe we should keep the box shut on these guys.
Now, we're going to test the convergence or divergence of some series by doing some reasoning similar to what we did when studying left-hand and right-hand sums for integrals.
Integral Test: Let f be a non-negative decreasing function on [c, ∞) where c is an integer. If the integral
converges, then the series
If the integral diverges, then the series also diverges.
Like we mentioned before, the integral test replaces grilled cheese with gremlins. They're hard to tame, but if you can do so, you can open the box safely. If you can't tame the gremlins, then keep that box shut.
The integral test works because, depending on how we draw the series, we can choose whether the rectangles will cover more or less area than the integral.
In the example with the harmonic series we drew the series as an overestimate. Since the integral diverged, we knew the series had to diverge.
If we have an integral that converges, we draw the series as an underestimate (right-hand sum) instead. Since a convergent integral describes a finite area, the smaller area covered by the rectangles must also be finite.
The last convergence tool we have is the comparison test. If all else fails, we should compare our Pandora's box to another one. If we look at the other one, and we decide the other one is bursting at the seems, we know it's safe to open ours up.
We studied improper integrals a while back, and we learned that, if f ≤ g on the interval (c, ∞), then
Here, when f and g are nonnegative, there's a smaller area under f and than under g. If the big area is finite, the smaller area must be finite too. If the small area is infinite, the bigger area must be infinite. If the other box is smaller and it's too dangerous to open, then ours is too. If is the other box is larger and it is safe to open, so is ours.
Since we can visualize a series as an area , we can use the same intuition to compare series. Suppose we have two series
0 ≤ an ≤ bn
for all n. Then the small area described by the series a is contained in the big area described by the series B.
This tells us two useful things:
Be Careful: As with improper integrals, we have to be careful about which way the comparisons go.
The tricky part, as with improper integrals, is finding the correct series to compare with. We can't compare gremlins to grilled cheese. We need something similar and easy to tell if the series converge or diverge. We like to use series of the form
whenever possible, since we can easily tell whether such series diverge or converge.
Be Careful: We can't use the comparison test if we can't find something to compare with. For example, we can't use the comparison test on
The only thing we can see to compare the term with is , but the inequality goes the wrong way:
As with most things in math, there are a few things that just can't fit nicely into the standard size boxes we try to to put them in. In the case of a few series, we need a Pandora's box that can open from 2 different ends.
Take a better look at what kind of beasts we're dealing with that we need a special box to constrain them. We can draw rectangles for series with negative terms the same way we drew left- and right-hand sums for integrals of functions with negative values. If a term is negative, we stick its rectangle under the horizontal axis.
Suppose is a series whose terms may be positive and/or negative. We can make another series by taking the absolute value of each term:
Visually, if the rectangles for look like this:
then to get the rectangles for we take any rectangles that were below the horizontal axis and flip them to lie on top of it instead:
If the original series converges but the series of absolute values doesn't, we say the original series converges conditionally.
These series are the Dr. Jekyll and Mr. Hyde of convergence. They are well-behaved on their own, but when you mix in a little potion with an absolute value symbol, they are out of control and diverge. If we try to put them in our Pandora's box, sometimes it's safe to open the box, and sometimes it's not.
Here's what conditional convergence looks like. Suppose has both positive and negative terms, and converges conditionally.
Since the limit of the partial sums converges, the positive and negative terms mostly cancel each other out as we go along:
Since converges conditionally, the series doesn't converge. This means the rectangles for cover an infinite area:
If the series of absolute values converges we say the original series converges absolutely. If the series of absolute values converges, it conveniently forces the original series to converge also. Put another way, if Mr. Hyde is well-behaved, so is Dr. Jekyll. The box is safe to open from either side.
Here's what absolute convergence looks like.
If converges, the region covered by all its rectangles is finite.
Any sub-region of a finite region is finite. That means the region covered by rectangles corresponding to positive terms of is finite.
And the region covered by rectangles corresponding to negative terms of is finite too.
The weighted area covered by the rectangles for must be the area above the axis minus the area below the axis:
This value is one finite number minus another, so it's finite, which means is finite. The series converges.
Practically speaking, this means if we can show converges, then we know that converges.
The alternating harmonic series converges conditionally but not absolutely, because
The geometric series converges absolutely because
converges, being a geometric series with |r| < 1.
We have an entire tool belt full of convergence tests to determine if series converge or diverge. The tests are as diverse as tools on a tool belt, too. The divergence test is a drill to bore holes into a series we suspect diverges, and the comparison test is a tape measure for us to compare to other series. We can even use the relationships of absolute and conditional convergence like a hacksaw to cut a series in half using the absolute value to check for absolute convergence. Tim "The Tool Man" Taylor would approve.
Now that we have all of the these fancy tools hanging around our waist, we need to now which tools to use when. It takes a bit of a mental jump to go from problems that say
"use the ratio test to determine if this series converges"
to problems that say
"determine if this series converges"
without giving any hints about which test(s) you should be using.
We presented the tests more-or-less in the order you should try them.
Just like woodworking or machining a widget, the best way to get better is to practice. Learning to use the right tools can be frustrating in any situation. We should keep that in mind, being grateful that these aren't real power tools. We can't cut a thumb off using the alternating series test.
There are multiple right ways to solve most of these problems. If you use a way we didn't mention, check with someone else to see if you found another correct way.