# Series Exercises

#### Series: This is the Sum That Doesn't End

If you've never heard "The Song that 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 suc...

#### Alternating Series

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 usu...

#### Convergence of Series

Now that we know how to bake an infinite brownie, we want to know what it tastes like. We're going to need to find a cow big enough to give us a glass of milk to match.In terms of series, this me...

#### 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 the...

#### Arithmetic Series

The arithmetic series is 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, fet...

#### 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...

#### 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 sp...

#### Decimal Expansion

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 ma...

#### Word Problems

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...

#### 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...

#### 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 co...

#### The Alternating Series Test

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 chees...

#### 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 hotdog...

#### The Integral Test

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...

#### 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 bursti...

#### 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...

#### 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 hol...

#### 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...

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