Testwiki:Reference desk/Archives/Mathematics/2018 June 17

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June 17

Convergence of Taylor expansion

I asked Wolfram Alpha to show me the Taylor expansion of x^.5 at 1 so I could check my math. But it included a region of convergence. Can someone tell me how to determine the region of convergence? RJFJR (talk) 23:45, 17 June 2018 (UTC)

Template:Ping For one-variable Taylor series, the Cauchy–Hadamard theorem provides a straightforward formulaic way to determine the radius of convergence. In this case, we have a Taylor series given by $\sqrt{x}=-\sum _{n=0}^{\mathrm{\infty }}\frac{(2n-3)!!}{n!(-2{)}^n}(x-1{)}^n=-\sum _{n=0}^{\mathrm{\infty }}\frac{(2n-3)!!}{(2n)!!(-1{)}^n}(x-1{)}^n$ from which we clearly see that the limit superior of the nth root of the absolute value of the nth term is 1 as $n\to \mathrm{\infty }$, yielding a radius of convergence of 1. This can perhaps more easily be demonstrated using the ratio test. Since we took the expansion about $x=1$, the result follows.--Jasper Deng (talk) 00:14, 18 June 2018 (UTC)

Thank you. RJFJR (talk) 03:09, 18 June 2018 (UTC)