The Limit Comparison Test

The comparison test lets us compare an unknown series to a simpler, known series. But it requires us to come up with inequalities that can be difficult to figure out. Worse, sometimes it just doesn’t quite work. For example, suppose we’re looking at the series

    \[ \sum_{n=2}^{\infty}\frac{1}{n^2-1}. \]

This series looks almost like a p-series with p=2, so we expect it to converge. However, if we try to use the comparison test to show that, we end up comparing n^2-1 with n^2. We have

    \[ n^2-1 < n^2, \]

so

    \[ \frac{1}{n^2-1} > \frac{1}{n^2}. \]

But that’s not the inequality we want! Using this inequality we can only show that our series has terms that are larger than the corresponding terms of a converging series. That doesn’t say anything about whether or not our series converges. So how can we proceed?
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The Comparison Test For Series

Sometimes we’re given a series that is very close to something we can deal with. For example, the series     is very close to the -series     In cases like this, we can use the comparison test to decide whether the series converges. First, remember that the second series above converges since it’s … Continue reading

The Harmonic Series

A special type of p-series is the harmonic series. This is a p-series with , in other words the series     Since p-series only converge when , the harmonic series diverges. Notice that the harmonic series is an example of a series whose terms tend to zero () but the series doesn’t converge. Remember … Continue reading

p-Series

Using the integral test, we can look at an important class of infinite series: the so-called p-series. These are series of the form     for some number . Here are some examples:     (a p-series with ),     (a p-series with ), and     (a p-series with ). Here’s the important … Continue reading

Absolute Maxima And Minima Problems

Finding the absolute maxima and minima of functions (the biggest and smallest values a function attains) is important for many real world calculus applications. For example, companies want to maximize their profits, engineers want to minimize the drag on an airplane, computer scientists want to get your youtube video from the server to your computer … Continue reading

The Integral Test For Series

One way we can determine whether or not a series converges is to view it as an area and see if the area is finite. Of course, we know how to find the area under a function (by integrating), so this suggests that integration might somehow be useful. That’s what’s involved in the integral test … Continue reading

Partial Fractions (Or, Partial Fraction Decomposition)

There are some situations when it’s useful to write a rational function (a quotient of polynomials) as a sum of simpler such functions. As an example, we can see by cross multiplying that     Partial fraction decomposition (often called just partial fractions) allows us to go backward, to figure out how to write as … Continue reading

Telescoping Series

Some series look complicated, but they’re really simple. Consider this one:     It’s pretty clear that every term except the 1 cancels, so the series should converge to 1. (In fact, it’s not too hard to show this. Can you write an expression for the partial sum ?) Telescoping series are similar to the … Continue reading

Geometric Series

The first type of series we usually learn about is called a geometric series. They’re so named because the ratio or successive terms is constant, but who cares about that. What do they look like? Here are two examples:     and     are both geometric series. In sum notation, we can write these … Continue reading

The Divergence Test

Now that we can find the sum of an infinite series when it converges, we’d like to be able to tell if a given series really does converge. Often it’s not easy to tell. But there’s one great test that can sometimes tell us when a series doesn’t converge. It’s called The Divergence Test