Radius and Interval of Convergence of Power Series

#10.13 Radius and Interval of Convergence of Power Series

Oooookay, that title definitely had a lot of buzzwords… namely, radius of convergence, interval of convergence, and power series. 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! 😉

#👊 What’s a Power Series?

A power series is a series of the form $\sum^{\infty}_{n=0}a_n(x-r)^n$, where n is a non-negative integer, $a_n$ is a sequence of real numbers, and r is a real number.

In this case, $a_n$ 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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