MATH-138-Fall-2016-Final.pdf

# A 5 x dy dx y x y 1 2016 b 5 dy dx 6 e 2 x y y 0 0

• Test Prep
• 14

This preview shows page 4 - 9 out of 14 pages.

(a) [5] x dy dx = y - x , y (1) = 2016 (b) [5] dy dx = 6 e 2 x - y , y (0) = 0 CROWDMARK MATH 138 Fall 2016 Final © 2016 University of Waterloo Please initial: Page 4 of 14

Subscribe to view the full document.

4. Consider the sequence ( a n ) defined by a 1 = 1 and a n +1 = 1 + 2 a n for n 1. (a) [3] Prove using induction that ( a n ) is increasing. (b) [3] Prove using induction that ( a n ) is bounded above. (c) [1] State the name of the theorem that guarantees the limit lim n →∞ a n exists. (d) [3] Determine lim n →∞ a n . CROWDMARK MATH 138 Fall 2016 Final © 2016 University of Waterloo Please initial: Page 5 of 14
5. (a) [3] State the definition of a series n =1 a n converging to a limit L . (b) [7] Prove that if a series n =1 a n is convergent, then lim n →∞ a n = 0 . CROWDMARK MATH 138 Fall 2016 Final © 2016 University of Waterloo Please initial: Page 6 of 14

Subscribe to view the full document.

6. Determine whether each the following series is absolutely convergent, conditionally convergent or divergent. Justify your answers. (a) [5] summationdisplay n =1 cos( n ) + sin( n ) + 1 n 2 (b) [5] summationdisplay n =1 ( - 1) n sin n CROWDMARK MATH 138 Fall 2016 Final © 2016 University of Waterloo Please initial: Page 7 of 14
(c) [5] summationdisplay n =1 5 n + 7 n 2 n + 4 n + 6 n (d) [5] summationdisplay n =1 ( - 1) n sin parenleftbigg 1 n parenrightbigg CROWDMARK MATH 138 Fall 2016 Final © 2016 University of Waterloo Please initial: Page 8 of 14

Subscribe to view the full document.

7. Determine the radius and interval of convergence for each of the following power series. Justify
You've reached the end of this preview.
• Winter '07
• Anoymous
• Mathematical analysis

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern