Section 14841 MWF 10 11am SEC 100 Math 1432 Dr Melahat Almus almus math uh edu http www math uh edu almus OFFICE HOURS at CASA MWF 1 2 pm COURSE WEBSITE http www math uh edu almus 1432 fall15 html Visit my website regularly for announcements and course material If you email me please mention the course 1432 in the subject line Bubble in PS ID and Popper Number very carefully If you make a bubbling mistake your scantron can t be saved in the system In that case you will not get credit for the popper even if you turned it in Check your CASA account for Quiz due dates Don t miss any online quizzes Be considerate of others in class Respect your friends and do not distract anyone during the lecture 1 POPPER Q Find the sum of the series if possible A 4 B 5 C 4 5 D 5 4 E Divergent A Q Find the sum of the series if possible 3 2 B 15 2 C 2 5 D 5 2 E Divergent Q Find the sum of the series if possible B 16 3 C 1 12 D 1 16 E None k 5 4 1 k 5 3k 0 k 2 k 1 2 2 k A 4 3 Q k 1 1 k 2 4 3 3 k 0 C Convergent D Divergent 2 Section 9 4 Tests for Convergence The Integral Test Comparison Tests In sections 9 4 and 9 5 we consider series with NON NEGATIVE terms Theorem 9 4 1 A series with non negative terms converges if and only if the sequence of partial sums is bounded Theorem 9 4 2 The Integral Test Let f be a continuous decreasing and positive function on 1 converges if and only if f k f x dx converges diverges if and only if f k f x dx diverges 1 1 1k 1k 3 Applications of the Integral Test Example The Harmonic Series k 1 1 k 1 1 2 1 3 1 4 4 Example The p Series with p 2 k 1 1 2 k 1 1 4 1 9 1 16 5 FACT The p Series k 1 1 p k 1 1 p 2 1 p 3 1 p 4 1 p k k 1 1 p k k 1 converges if 1p diverges if 0 1p Example 1 k k 1 Example 3 2 5 k k 1 6 Example Determine whether the series is convergent or divergent 2 k 1 2 k ln k 7 Theorem 9 4 5 Basic Comparison Test If ka and 0 kb and 0 If kb converges and 0 a k b k then ka converges If ka diverges and 0 a k b k then kb diverges 1 2 8 So Let ka and 0 kb 0 If A diverges and B A what happens If A converges and B A what happens If A converges and B A what happens If A diverges and B A what happens 9 Examples k 1 1 5 k 2 2 1 k 2 2 k 1 2 k 10 2 k 1 3 k 4 k 2k k k 2 5 k k 2 1 4 11 1 k k 2 lnk k 4k 12 Popper Choose C for convergent D for divergent Q 5 k 2 3 1 k k 1 7 1 k k 1 1 5 k k 1 k 1 3 2 k k 1 k 1 k 1 3 1 5 k k 1 2 1 Q Q Q Q Q Q 13