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MAT420 Exam 2 Solutions, Exams of Mathematics

State University of New York Polytechnic - Utica-RomeMathematics

Solutions to exam 2 of mat420, due on november 13, 2007. It includes answers to questions about series convergence, limits, complex functions, and integrals.

Typology: Exams

Pre 2010

Uploaded on 08/09/2009

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MAT420 Exam 2 Due November 13, 2007
Prof. Thistleton
Please write all answers in the space provided.
1. Do the following series converge? Justify your answer.
(a) P∞
n=1 i
n+1
n!
(b) P∞
n=1 1+i
n2
(c) P∞
n=1
(1−i)n
2n
(d) P∞
n=1
(1+i)n
n2
(e) P∞
n=1
in
n!
1
pf3
pf4
pf5

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MAT420Prof. ThistletonPlease write all answers in the space provided. Exam 2 Due November 13, 2007

  1. Do the following series converge? Justify your answer.(a) ∑∞ n=1 ( (^) ni + (^) n (^1)! ) (b) ∑∞ n=1 1+n 2 i (c) ∑∞ n=1(1− 2 ni)n (d) ∑∞ n=1(1+ n 2 i)n (e) ∑∞ n=1 i nn!
  1. Where do the following series converge?(a) ∑∞ n=1(2 + i)nzn

(b) ∑∞ n=1 (^) (5+12^1 i)n zn (c) ∑∞ n=1(z− 2 n1)n

  1. Find the limits:(a) (^) nlim→∞ in (^) n+ 1!

(b) lim sup n→∞ |3 + in|

  1. Using the software of your choice, plot exp(z) for z(t) = t + (2π + t)i where − 4 ≤ t ≤ 4.
  2. Calculate Log(√2 + i).
  3. Find z such that Log(z + i) = iπ.
  4. Find z such that cos(z) = 2.
  5. Evaluate ∫^02 (t + i)^2 dt
  1. Consider(a) Fill in the following table. 1 + 0 (^) z∫k C (^). 0 fi ( z)∆dzz (^) kwhere (^) 1 + 0c fk (^) .( 1 zi) = z^2 fand (ck (^) )z (t) = 1 + (^) fit ( (^) cfor 0k)∆z ≤k t ≤ 1. 1 + 0 1 + 0 1 + 0 1 + 0 1 + 1..... 24680 iiiii 1 + 01 + 01 + 01 + 0.... 3579 iiii ∑ (^) = (b) Now evaluate the same integral analytically.
  2. ∫ C z^2 dz for the top half of C 1 + (0).
  3. ∫ C z^2 dz for z(t) = 1 − 2 t, 0 ≤ t ≤ 1.
  4. ∫ C+ 3 (0)z^ − (^2) +1^2 i dz.