Georgia Tech - MATH 1551 - Class Notes - Week 8

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Unformatted text preview: Concepts HREE TANGIS WE HAVE DISCUSSED SO FAR THIS SEMESTER numer cat mtegration (16P urrkuatims (saisha FTC definite integrals Improry intearat V sub rriemann sums - net area indev tuve - aucumulation indef. intitecho- integrals iques undoing denivitives ant- derivanves fithm differertial Cquations applications- impioper integrals - A1III Before now we required a finite domain + range improper integrats are calculated as limits there is convergence, the limit is finite Discontinuit'es also cause improper integrals If the limit diverges, the limit IS infiinite pireet compan'son Test and Limit Compan'son test test for comergenu or divergen le it's not enough just to test endpolrits |Limit wmpan son if lim tlx- L,0 L s os TUST Xogo | ther, 5* (x)dy and 5* glx) dx both converge or botn diverge Direct Companisan let t and g be continuous Test on fa, c) with Dif(x)x) Then, for all x2 a. OS Gx) dx converges if / glx)dx does OS (x) dx diverges if f* fix) dx does PRACTICE O lim SEAL110001 SS NOTES 10,- Seauenug,fries Baja sequences e.g. 1, 1 1 1 senes eg. 1, 142, 1+1+1 Pomer seres eg: 1, 14 15 *Any smooth function cam be approximatice well locally loy a power series. (smooth means denivihives van loe taken annjwmere ) * A Sequence is an infinite list of numbers - eaim number in a segwenee isa tem - term subscripts are calle their index - fibonacci Sequence * recursive tg-defined sequence ex= An: an-tan-2 * poup per bound of a sequence E43 is a number M such that A EM for all n * There can lee an infinite amount of prek bunda + least upper bound is the smallest un HA (ower bound of a sequence {An3 is a num laer M Suin tat M for all N. * Greatest lower bound is the largest lower bound THEOREM A seavenie wnverges if it is monotone and bounded (VB+LB exist * * sequence is ononotone if it t weakly increasin or weakly decreasing, f.e. 1,2,2,3,3,3. * for tne theorem, it doesn't have to start out mond.

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