Category Archives: series
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 This series looks almost like a p-series with … Continue reading
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
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
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
Where To Start The Index Of A Sum
When we look at infinite sums, they almost always start the index at 0 or 1. For example, a geometric series looks like and the number is equal to So do sums have to start with 0 or 1? No! They can start wherever we like. It just happens that 0 … Continue reading
How To Find The Sum Of An Infinite Series
Now that we know what a series is we need to figure out how to assign a sum to them. The trick is this: we turn the series into a sequence, and if that sequence converges we’ll call its limit the sum of the series. In detail, given a series define a sequence … Continue reading