MATH138 Study Guide - Final Guide: Telephone Numbers In The United Kingdom, Ratio Test, Squeeze Theorem

• School: University of Waterloo
• Department: Mathematics
• Course: MATH138
• Professor: All

Document Summary

We need to prove lim , () = 0, then we can say () = . ! > lim ||+1 = 0 < 1 so the series converges. = 0, so , 0 by squeeze theorem. And we can make ad large as we want. Ex: find taylor series about = 6. ,() = || () = 0 (6) = 6. () = 1 > 0 (6) = 1. This doesn"t contradict the convergence theorem since (0) one. 0 = 0 (no exists for ()() ) Ex: cos: find maclaurin series for cos () = cos (0) = 1. =0 (2)! lim + 1 = lim ( 1)+1 2+2 (2 + 2)! = 0 (2 + 2)(2 + 1) Since ()() 1 , , so it holds by the convergence theorem. = 0 sin = ( 1) 2+1 sin(0) = 0 + (2+1)! Ex: find taylor series centered at = 3 for () =