10.13 Radius and Interval of Convergence of Power Series
4 min read•june 18, 2024
10.13 Radius and Interval of Convergence of Power Series
Oooookay, that title definitely had a lot of buzzwords… namely, radius of convergence, , and . You haven’t seen them in any of the previous study guides, either, so they’re definitely new to you. Let’s define them one by one! 😉
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This is an AP Calculus BC topic only! If you are taking Calculus AB, you can skip this material. If you’re taking AP Calculus BC, here you go! ⬇️
👊 What’s a Power Series?
A powerseries is a series of the form ∑n=0∞an(x−r)n, where n is a non-negative integer, an is a sequence of real numbers, and r is a real number.
In this case, an can be any sequence, most of which you’ve already seen in previous study guides! r refers to where we center our power series function (e.g., a “power series centered at x = 3” will give us (x - 3) at that part of the series).
🔵 Radius and Interval of Convergence
One of the questions we have about power series approximations of functions is where the approximation is valid, or in other words, where the power series converges. For a given x, we can find the radius, and then the interval of convergence for a power series.
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If a power series converges, it either converges at a single point or has an interval of convergence. The ratio test can be used to determine the radius of convergence of a power series.
For a centered at x = r, the only place where we are entirely sure that it converges to is at x = r, but we can expand this to a greater range using our knowledge of the . Let’s make an example to demonstrate this!
One last thing: we need to test these —namely, the two extreme points of a line segment or interval—by plugging the values into the original series to see if they are included in our solution or not:
From our previous encounter with the alternate harmonic series above, we can say that the series converges at x = 15/8. In other words, 15/8 is included in our interval of convergence. What about x = 17/8?
Another familiar face: the harmonic series! We can, thus, say that the series diverges at x = 17/8. In other words, 17/8 is not included in our interval of convergence.
Altogether, our interval of convergence is [815,817) or 815≤x<817.
⭐ Summing Up Power Series
To summarize what we introduced above:
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Power Series
Power series allow us to represent associated functions on an appropriate interval!
If a power series converges, it either converges at a single point or has an interval of convergence.
The ratio test can be used to determine the radius of convergence of a power series.
The radius of convergence of a power series can be used to identify an open interval on which the series converges, but it is necessary to test both endpoints of the interval to determine the interval of convergence.
To make things easier for you, here’s a quick guide on what you should do when you encounter a power series problem that asks you to find the radius & interval of convergence:
Apply the ratio test!
Use the ratio test to find your radius of convergence and endpoints.
Plug endpoints back into your original series to see if they are included in the solution or not… this’ll help you finalize your interval of convergence!
If the endpoint series converges, that endpoint is included.
If the endpoint series diverges, that endpoint is not included.
Use your “series and tests” (study guides 10.3 to 10.9) toolkit!
That’s it! While the journey to the answer seems long and arduous, you’ll notice that the building blocks from earlier study guides and math courses like the p-series test, the ratio test, and even absolute values all come together in the concept of power series. As always, mastery comes with practice, and becoming an expert at this topic will help you brush up on the prior concepts in this unit as well.