Infinite series can represent functions.
Here are four basic power series (centered at zero) that every mathematician knows.
Next to each of the series, we list an interval which will correspond to the domain of the series. Using power series we can “read-off” properties of functions. Here are some examples.
- We can easily see that , , and .
- Since every power of in the power series for sine is odd, we can see that sine is an odd function. Likewise, since every power of in the power series for cosine is even, we can see cosine is an even function.
- Limits like are “easy” to compute, since they can be rewritten as follows.
- Power series give us methods to actually compute values for these functions.
You may have noticed a small caveat above. The caveat is the “,” which we referred to as something like the domain of the function. This restriction is required because if our formula is true, then for any number , provided that , we have and the expression on the right-hand side of the equation above is a geometric series! As we’ve learned, geometric series only converge when the common ratio (in this case ) is between and noninclusive. If we look at a graph of along with a graph of we see
Our next theorem tells us what possible scenarios we could encounter when investigating convergence of power series.
- The series converges only at .
- There is an such that the series converges for all in and diverges for all and .
- The series converges for all .
Power series are both similar to and different from the series we’ve previously studied. When we fix some value for , we are working with the sort of series we’ve already studied - a series of numbers. In this way, we can use all of our previous tools for working with series. We can also let be a variable, and consider our power series as a function. Because power series can define functions, we no longer exclusively talk about convergence at a point, instead we talk about the radius and interval of convergence.
- If a power series converges absolutely for all , then its radius of convergence is said to be and the interval of convergence is .
- If a power series converges absolutely for all in and diverges for all and , then its radius of convergence is said to be and the interval of convergence is one of the following:
- If a power series converges only at one value , then its radius of convergence is said to be and the series does not have an interval of convergence.
In the previous definition, the interval of convergence depends on the series. We must separately consider the behavior of a power series at the endpoints of its interval of convergence. In other words, we plug in values for , and consider the series as a series of numbers!
How do we check for radius of convergence? Two old friends can come to the rescue: the ratio and root tests.
Now, for any fixed value of , we have that since we recall that is a constant in this limit and its value does not affect the value of the limit. Hence, the radius of convergence for is , and the interval of convergence is .
While the ratio and root test are good for determining the radius of convergence of a power series, they are useless for determining convergence at the end-points of the interval. Let’s see an example:
Using logarithms and L’Hôpital’s rule, we can show that Hence The root test gives us convergence when this limit is between and . In other words, the series converges absolutely when
However, and so adding to all sides of the inequality, we need such that Since our power series is centered at , the radius of convergence is . However, the root test (and ratio test) is inconclusive at the end points and . For this, we need to investigate separately the following two series, found by plugging in and . For the first, where , note that
This is the alternating harmonic series, which we know converges. So our power series converges at . For the second, where , note that
This is the harmonic series, which we know diverges. So our power series diverges at . Hence the interval of convergence for must include everything between and , as well as , but does not include . In other words, the interval of convergence is .
Let’s work through an example of a power series that only converges at a single point.
With the basic power series above, we can produce new power series via algebraic manipulation.
converge absolutely for , and let be a continuous function.
- for .
- for .
- for .
In our first example we will derive Euler’s famous formula where is the number .
Euler’s formula allows us to produce (by setting ) the amazing identity: This identity combines the fundamental constants, , , , and , along with the fundamental operations of addition, multiplication, and exponentiation!
- is continuous and differentiable on .
- , with radius of convergence .
- , with radius of convergence .
- The theorem states that differentiation and integration do not change the radius of convergence. It does not state anything about the interval of convergence. They are not always the same. Check the endpoints!
- Notice how the summation for starts with . This is because the derivative of the constant term of is .
- Differentiation and integration are simply calculated term-by-term using the power rule.
Let’s see an example.