midterm1-soln - AMS-510 Analytical Method for AMS Midterm...

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 DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: AMS-510 Analytical Method for AMS Midterm Exam 1 Do all the problems, six right will get you full credit (1). Find the following limits (a). lim n →∞ n + 4 n n (b). lim n →∞ 1 2 n + 1 3 n 1 n (c). lim x → cos 2 x- 1 x 4- x 2 (d). lim n →∞ 1 n 1 + 1 n 2 + 1 + 2 n 2 + ··· 1 + 2 n n 2 ! Solution: (a). L = lim n →∞ n + 4 n n = lim n →∞ 1 + 4 n n Let n = 4 x , L = lim n →∞ 1 + 1 x 4 x = lim n →∞ 1 + 1 x x 4 = e 4 (b). lim n →∞ 1 2 n 1 n ≤ lim n →∞ 1 2 n + 1 3 n 1 n ≤ lim n →∞ 1 2 n + 1 2 n 1 n LHS = 1 2 , RHS = lim n →∞ 2 1 2 n 1 n = 1 2 lim n →∞ 2 1 n = 1 2 Therefore the limit is 1 2 . 1 (c). L = lim x → cos 2 x- 1 x 4- x 2 Using L’Hospital’s rule, we have L = lim x →- 2 sin 2 x 4 x 3- 2 x = lim x →- 4 cos 2 x 12 x 2- 2 =- 4- 2 = 2 (d). L = lim n →∞ 1 n 1 + 1 n 2 + 1 + 2 n 2 + ··· 1 + 2 n n 2 ! This the limit of Riemann sum L = Z 3 1 x 2 dx = 1 3 x 3 | 3 1 = 9- 1 3 = 26 3 (2). Given function f ( x ) = | x | 3 , find:...
View Full Document

This note was uploaded on 02/28/2012 for the course AMS 510 taught by Professor Feinberg,e during the Fall '08 term at SUNY Stony Brook.

Page1 / 5

midterm1-soln - AMS-510 Analytical Method for AMS Midterm...

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