FinalMath2JSummer11_solutions

FinalMath2JSummer11_solutions - Math 2J Summer 11...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Math 2J, Summer 11 Instructor : Yunho Kim The Final Exam Date : Sep. 7th UID Name Class Lec. B, 9:00am - 10:50am Number Score 1. a) b) 2. a) b) 3. 4. 5. 6. 7. Total Score / 100 1 Problem 1. Determine if the following series are convergent. Justify your answers . And if it is convergent, then find the value of the series. a. (10pts) ∞ summationdisplay n =1 a n , where a n = 1 n 2 + 5 n + 6 Solution. By the partial fractions method, we know that ∞ summationdisplay n =1 1 n 2 + 5 n + 6 = ∞ summationdisplay n =1 parenleftBig 1 n + 2- 1 n + 3 parenrightBig = lim k →∞ k summationdisplay n =1 parenleftBig 1 n + 2- 1 n + 3 parenrightBig Notice that the k th partial sum is k summationdisplay n =1 parenleftBig 1 n + 2- 1 n + 3 parenrightBig = parenleftBig 1 3- a3 a3 a3 1 4 parenrightBig + parenleftBig a3 a3 a3 1 4- a3 a3 a3 1 5 parenrightBig + ··· + parenleftBig a0 a0 a0 1 k + 2- 1 k + 3 parenrightBig = 1 3- 1 k + 3 . Therefore, ∞ summationdisplay n =1 1 n 2 + 5 n + 6 = lim k →∞ parenleftBig 1 3- 1 k + 3 parenrightBig = 1 3 . b. (10pts) ∞ summationdisplay n =1 a n , where a n = 2 n + (- 3) n 4 n Solution. Notice that ∞ summationdisplay n =1 2 n + (- 3) n 4 n = ∞ summationdisplay n =1 2 n 4 n + ∞ summationdisplay n =1 (- 3) n 4 n = ∞ summationdisplay n =1 parenleftBig 1 2 parenrightBig n + ∞ summationdisplay n =1 parenleftBig- 3 4 parenrightBig n and both infinite series are geometric series. Therefore, ∞ summationdisplay n =1 2 n + (- 3) n 4 n = ∞ summationdisplay n =1 parenleftBig 1 2 parenrightBig n + ∞ summationdisplay n =1 parenleftBig...
View Full Document

{[ snackBarMessage ]}

Page1 / 8

FinalMath2JSummer11_solutions - Math 2J Summer 11...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online