Infinite Series

Infinite Series

In all of our previous examples, the series all had a finite number of terms, i.e. the counters all had a starting number and an ending number. If we take off the upper number and let the series keep going indefinitely, we get what's called an "infinite series". For example

$$\sum_{i=1}^\infty \frac{1}{i} = \frac{1}{1}+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+...$$ $$\sum_{i=0}^\infty 2^i = 1+4+6+9+16+...$$ $$\sum_{i=0}^\infty \frac{1}{2^i} = \frac{1}{1} + \frac{1}{4} + \frac{1}{8} + ...$$

At this point, you might be thinking, "How can we talk about adding up infinitely many numbers? That can't be possible." That's a valid point and the first thing I should clarify is that when we talk about "the sum of an infinite series", we aren't talking about adding up all the numbers like we did with finite series. To see how we define this kind of sum, take a look at the last series above,

This is just the geometric series with r = 1/2 that we saw in the last section. The table below has the results that we get from adding up the first few terms of the series

The diagrams in the right hand column illustrate what's going on as the number of terms increases. Notice how every time we add a term, the total gets a little bit closer to 2 but never goes past it. That's what we mean when we say that an infinite series equals a number: As the value of i gets bigger, the sum gets really close to the number but never becomes greater than the number.

If all this talk about "really close to" seems kind of vague, you're right. A precise definition of what we mean by "close" belongs in a calculus class so we're going to have to put it off for later. For now, what you should understand is that the "sum" of an infinite sequence is the smallest upper limit on the "partial sums", i.e. the sums that you get from adding up the beginning part of of the series.

Infinite Geometric Series

Finding that "smallest upper limit" for a specific series can be a tricky business. In the special case of our geometric series, there's a simple formula that you can use.

Keep in mind that our formula has an important limitation: |r| < 1. What happens if |r| ≥ 1? In those cases, the series doesn't have a sum.² This is an important point that you need to be aware of: Not all infinite series have a sum or "converge" to a value.

Approximations with Infinite Series

Suppose I asked you to find the value of e^1.8. You'd probably just take out your calculator, press a few buttons and give me the answer. That works great today but what would you do without your calculator? There's a whole branch of math called approximation theory that studies how you can take hard to calculate functions like f(x) = e^x and approximate them using functions that are easier to work with. Many of these approximations end up being infinite series. For example,

$$f(x)=e^x = \sum_{i=0}^{\infty} \frac{x^i){i!}$$

What makes a formula like this useful is that it gives you a way to calculate values for the function that just involve basic arithmetic, i.e. multiplication, division, addition and subtraction.

When we were looking at the formula for an infinite geometric series, it had a limitation. It only worked for values of r that were between -1 and 1. These approximation series, called Taylor series after Brook Taylor, an 18^th century mathematician who came up with a formula for deriving them, also often have a limited range in which they're accurate.

Exercises

² If it bothers you to talk about "a sum" when this isn't really the sum of a list of numbers, we also use the word "converge" in these situations, e.g. we could also say that the series in Example 1 converges to 9/2.

³ What you're seeing here is a common problem with Taylor series. Even for a value where the series converges, the number of terms that it takes to get a good result can be different for different values of x.