Ep 30: The Magic Number That Decides Where a Series Converges | Radius of Convergence

Last episode we discovered that smooth functions hide infinite polynomials inside them --- the Taylor series. But we left a question dangling: for which x does the series actually equal the function? e^x converges everywhere. 1/(1-x) only for |x| less than 1. Some series, like \sum n!\, x^n, converge only at a single point. Today we give that question a precise answer. Every power series \sum c_n (x - a)^n has a \emph{radius of convergence} R \in [0, \infty]. Inside the interval (a - R, a + R) the series converges absolutely. Outside, it diverges. On the boundary -- the two endpoints -- the behavior depends on the series and has to be checked by hand. We'll prove this dichotomy, compute R for half a dozen series using the ratio test, see how to differentiate and integrate a power series term-by-term inside its radius, and meet the real analytic functions --- the smooth functions that are honestly equal to their Taylor series. We'll also meet the famous counterexample: e^{-1/x^2}, smooth at zero but with a Maclaurin series that is identically zero. 📚 Continues in: → 'Functions of Several Variables and the 3D Coordinate System' --- power series end one-variable calculus. Next we step into multiple variables, surfaces in space, and a whole new geometry. 🎓 Best for: Calculus 1, 2, and 3 students, AP Calculus AB / BC, anyone reviewing before an exam, and self-learners who want a single coherent course instead of a pile of disconnected YouTube videos. — QED Realized — Clean explanations of the ideas that take a real lecture to land. 🔔 Subscribe so you don't miss the next one:    / @qedrealized   #Calculus #Math #Mathematics #STEM #PowerSeries #RadiusOfConvergence #Calculus2 #IntegralCalculus #RealAnalysis — Tags (for search) — power series tutorial, radius of convergence tutorial, interval of convergence step by step, ratio test for power series, root test for power series, cauchy hadamard formula, term by term differentiation of power series, term by term integration of power series, deriving ln(1+x) from geometric series, real analytic functions, smooth but not analytic, e^(-1/x^2) counterexample, taylor series radius of convergence, calculus 2 power series, AP Calculus BC power series, MIT power series, cauchy 1820 convergence theory, weierstrass real analytic, find radius of convergence ratio test, endpoint testing power series ⏱ Chapters 0:00 Where does an infinite polynomial actually live? 1:14 \sum c_n (x - a)^n 2:35 Convergence at one point forces convergence on a ball 4:07 R \in [0, \infty] exists; trichotomy 5:12 R = \lim |c_n / c_{n+1}| 6:39 R = \infty --- the exponential 7:45 R = 1 --- endpoints differ 9:09 R = 0 --- the lonely series 10:22 \sum (x - 3)^n / 2^n 11:39 Open ball around a, endpoints maybe included 12:33 Inside R, you can do calculus on the series 13:57 Integrate \sum (-x)^n to get \ln 15:34 Smooth functions that equal their Taylor series 16:58 Smooth but \emph{not} analytic 18:43 Convergence theory and analytic function theory 20:02 Recap 21:42 Outro 22:31 Thanks for watching