Examples of Convergent and Divergent Series

####Background

If you need a primer on series in general, check out this explanation. It gives an introduction to what a series is and dips a bit into the ideas of convergent and divergent series, so it's a good place to start. In these lessons, you will cover sum of an infinite series and sigma notation (finding the sum). It might even be helpful to review our lessons on repeating decimals and rational numbers.

A convergent series is a series whose partial sums tend to a specific number, also called a limit. A divergent series is a series whose partial sums, by contrast, don't approach a limit. Divergent series typically go to $\infty$, go to $-\infty$, or don't approach one specific number.

An easy example of a convergent series is $$\sum\limits_{n=1}^{\infty}\frac{1}{2^n} = \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \cdots$$ The partial sums look like $$\frac{1}{2}, \frac{3}{4}, \frac{7}{8}, \frac{15}{16}, \cdots$$ and we can see that they get closer and closer to $1$. The first partial sum is $\frac{1}{2}$ away, the second $\frac{1}{4}$ away, and so on and so forth until it is infinitely close to $1$. This means that the above series converges to a limit at $1$!

(The ! does not indicate a factorial, although in the case of $1! = 1$ it wouldn't matter. Limits are just exciting.)

The convergence of different series is a very relevant topic in higher math and many significant mathematical discoveries revolve around different series, but for the purposes of Algebra 2 we can mainly concern ourselves with whether or not geometric series will converge.

Given a geometric series with common ratio $r$, it will converge whenever $|r| < 1$. This should make intuitive sense because if we had a geometric series with a common ratio of say, $2$, we would have something that looks like $$\sum\limits_{n=0}^{\infty}2^n = 1 + 2 + 4 + 8 + 16 + \cdots$$ Clearly the numbers we are adding to the sum get larger and larger, so the sum would never slow down and approach a specific limit. This is an example of a divergent series.

Similarly, if the common ratio is $-2$, the series $$\sum\limits_{n=0}^{\infty}(-2)^n = 1 - 2 + 4 - 8 + 16 - 32 + \cdots$$ will also diverge. A way to better see this is $$\begin{align} \sum\limits_{n=0}^{\infty}(-2)^n &= (1-2)+(4-8)+(16-32)+ \cdots \\ &= -1 - 4 - 16 - \cdots \\ &= -\sum\limits_{n=0}^{\infty}4^n \end{align}$$ which is clearly divergent.

Now that you've been introduced to convergence and divergence, feel free to try out some of the problems in this topic. Have fun!

Math in Music

Now that you've seen the basics of series we are going to dive a little bit deeper into the details by thinking about the musical and mathematical concept known as the harmonic series.

In music, all harmony is based off of ratios. Originally the pitch of notes was described as divisions of a single string. If you had a string that sounds at a $C$, you could cut the string in half, and both of the parts would sound at a $C$ one octave higher.

Notice that the difference in size decreases as we go further into the series. This is also visible on a musical staff.

If you play a $C$ on an instrument, you actually hear all of the notes! The interesting part about the harmonic (or overtone) series in music is that they help make up the characteristic sound of an instrument. For example, part of the sound of a flute is made up by very high amounts of higher overtones. This is why a flute sounds so bright! The overtone notes don't overpower the note being played, a $C$ will sound like a $C$, even if there is a lot of sound coming from the $E$ overtone that is way above it.

In math, this means we now have a framework to think about the harmonic series and origin of the name, but it doesn't actually explain anything in terms of math. In summation notation, the harmonic series can be written as the following:

$$\sum\limits_{n=1}^{\infty}\frac{1}{n}= 1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}...$$

As the series continues, the terms get closer together similar to the notes of the musical harmonic series. This is awfully similar to the geometric series:

$$\sum\limits_{n=1}^{\infty}\frac{1}{2^n} = \frac{1}{2}+\frac{1}{4}+\frac{1}{8}...$$

which converges to $1$. As explained in the explanation on convergent series, this series converges because its common ratio has an absolute value less than $1$ ($|r|<1$). We can compare this to our harmonic series.

The harmonic series has no common ratio, so we can't automatically assume that it is convergent. Let's take a look at some partial sums to help us determine convergence or divergence.

$$\begin{align} \sum\limits_{n=1}^{4}\frac{1}{n}&= 1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}=2.083\\ \sum\limits_{n=1}^{5}\frac{1}{n}&= 2.283\\ \sum\limits_{n=1}^{10}\frac{1}{n}&= 2.929\\ \sum\limits_{n=1}^{30}\frac{1}{n}&= 3.994\\ \end{align}$$

Based off of our partial sums, there is no single value that the series converges towards. Every time a new term is added to the series, the sum slowly approaches infinity. Eventually the new terms added will be very small that the beginning digits will be the same (think of $10.15$ versus $10.19$) but they won't be small enough to converge towards one number.

We can prove this further with one more in depth explanation, but it can be a little tricky to think about. Let's list out the terms of our harmonic series: $$1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}+\frac{1}{7}+\frac{1}{8}...$$ We can compare this to a similar series. What we're going to do is round each one of our terms to a greater power of $2$: $$1+\frac{1}{2}+\frac{1}{\color{#8925AE}{4}}+\frac{1}{4}+\frac{1}{\color{#8925AE}{8}}+\frac{1}{\color{#8925AE}{8}}+\frac{1}{\color{#8925AE}{8}}+\frac{1}{8}...$$ This new series can be grouped into sections that equal $\frac{1}{2}$: $$1+\frac{1}{2}+\left(\frac{1}{\color{#8925AE}{4}}+\frac{1}{4}\right)+\left(\frac{1}{\color{#8925AE}{8}}+\frac{1}{\color{#8925AE}{8}}+\frac{1}{\color{#8925AE}{8}}+\frac{1}{8}\right)...$$ After simplifying our new groupings, we get the following series: $$1+\frac{1}{2}+\frac{1}{2}+\frac{1}{2}...$$ This series diverges as $\frac{1}{2}$ is added every single time. There is no upper bound to this and it continues onto infinity. Each section of our original series, $\sum\limits_{n=1}^{\infty}\frac{1}{n}$, is greater than the groupings equivalent to $\frac{1}{2}$. $$\begin{align} \frac{1}{3}+\frac{1}{4} &> \frac{1}{4} + \frac{1}{4}\\ \frac{1}{5}+\frac{1}{6}+\frac{1}{7}+\frac{1}{8}&>\frac{1}{8}+\frac{1}{8}+\frac{1}{8}+\frac{1}{8} \end{align}$$

Our series of $\sum\limits_{n=1}^{\infty}\frac{1}{n}$ must be divergent as at any term it is greater than or equal to the compared series.

Examples of Convergent and Divergent Series

Not every infinite series behaves the same way. Some infinite series approach a fixed value, while others continue growing or changing forever. This is why mathematicians classify infinite series as either convergent or divergent. We'll learn about both in this lesson!

Convergent Series

A convergent series is an infinite series whose partial sums get closer and closer to a single number. Even though the series has infinitely many terms, its total approaches a finite value called a limit.

For example,

$$1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + ...$$

Each term is half the previous one, so the terms become smaller and smaller. The partial sums get closer and closer to $2$, so this is a convergent series.

Let's look at another example!

$$1 + \frac{1}{4} + \frac{1}{16} + \frac{1}{64} + ...$$

Here, the terms decrease even more quickly. As a result, the series also converges to a finite sum.

####Divergent Series

A divergent series is an infinite series whose partial sums do not approach a single value. Instead, the total may increase without bound or continue changing forever.

For example,

$$1 + 1 + 1 + 1 + ...$$

This is a divergent series. Since each term is $1$, the total keeps increasing and never approaches a finite number.

Let's look at another example!

$$1 + 2 + 4 + 8 + ...$$

Each term is twice as large as the previous one, so the total grows larger and larger without a limit.

When deciding whether a series converges or diverges, look at how the terms behave. If the terms become smaller in the right way, the series may converge. If the terms stay the same size or grow larger, the series will diverge.

Learning to recognize these patterns will help you analyze many different types of infinite series. In later lessons, you'll use tests such as the Geometric Series Test and the Nth Term Test for Divergence to determine whether a series converges or diverges without having to calculate every partial sum.

Convergent and Divergent Series

In this lesson by the Organic Chemistry Tutor, you will learn the basics of both convergent and divergent series.