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Series: This is the Sum That Doesn't End
If you've never heard "The Song the Doesn't End", don't follow this link, we dare you.Never-ending series are much less annoying and much more useful. We promise.A series is a sum of numbers such a...
An alternating series is a series whose terms alternate between positive and negative like a light switch being flipped on and off. Just like alternating sequences, the terms of such a series usual...
Convergence of Series
Now that we know how to bake an infinite brownie, we want to know what it tastes like. We are going to need to find a cow big enough to give us a glass of milk to match.In terms of series, this mea...
Properties of Series
A Match Made in Math-HeavenIf series are peanut butter, then integrals are jelly. You could put one or the other on a couple slices of bread and have a satisfying sandwich. You could also put them...
The arithmetic series is the one of the simplest series we can come up with. In terms of common restaurant menu items, the arithmetic series is a burger. Although it many be spiced up with bacon, f...
Finite Geometric Series
It doesn't matter where the first term of a sequence begins. Much the same it doesn't matter too much where the first term of a geometric series begins. For the sake of making Sigma notation tidy a...
Infinite Geometric Series
Imagine the infinite ways you can top a hotdog. Relish looks kind of bland after you consider what strawberry sauce and sprinkles could do to a hotdog. Infinite geometric series, which are just spe...
We know all we need to know about geometric series. As a nifty bonus, we can use geometric series to better understand infinite repeating decimals. This is something you can rub in your former math...
Since arithmetic and geometric series are common menu items that everyone loves, they show up a lot in word problems. Unlike hotdogs and burgers, the word problems usually provide some individual q...
Visualization of Series
We've already mentioned that series and integrals are much alike. The integral is the series' brother from another mother. Before we put on our x-ray glasses and take a closer look at convergence a...
The Divergence Test
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 conv...
The Alternating Series Test
The first tool in our arsenal our 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...
The Ratio Test
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....
The Integral Test
We already know that series and integrals share some similar properties. We are going to show that, by replacing the series with an equivalent integral, we can determine if the series converges.In...
The Comparison Test
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...
Absolute Convergence vs. Conditional Convergence
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 o...
Summary of Tests
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...
Taylor and Maclaurin Series
Right when we thought we had seen it all, it's time to take a look at the big boys: Taylor and Maclaurin. We like old Brook and Colin, they made calculus class just a little bit easier—at least w...
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