MATH 2283 Sample Exam Explain all arguments clearly and concisely.

• 1). Let P, Q, and R be mathematical propositions. Compose a truth table for the proposition {not(P or Q)} and R.
• 2) Which of the following is true and which is false. For every false statement give a counterexample. Let A subset of R, the reals.
• a) If A is bounded then A is bounded above.
• b) If A is bounded above, then it is bounded below.
• c) If A is bounded above then it has a maximum element
• d) If A is bounded above then maxA = lubA.
• 3) Let T = {(0, 1) union {2 - 1/n | n a natural number}. Find lubT, glbT, maxT, minT , or indicate they do not exist.
• 4). Use the epsilon-N definition for convergence of a sequence to prove that the sequence 5n-2/4n+3 converges and has limit equal to 5/4.
• 5). Let pi be usual real number pi. Consider all real numbers U = {pi/n; n a natural number}. Given epsilon > 0, show there is some u in U such that 0 < u < epsilon.

MATH 2283 Sample Exam II Explain all arguments clearly and concisely.

• 1) Does the sequence (a_n) converge or diverge where a_n = (n^3 + 2)/(2^n +7). If it converges, give the limit. If it diverges, give a short explanation.
• 2) Does the series (not sequence!) with general term 1/(n^2 + 4) converge or diverge? The limits of summation are n=1 to n=\infty.
• 3) Suppose (a_n) is a monotonically decreasing sequence. Let K be a lower bound for the set of elements {a_n}. Let A=glb {a_n}. Prove lim a_n = A as n goes to \infty.
• 4) Give an example of a nested decreasing sequence of open intervals having empty intersection.
• 5)Does the sequence (sqrt{n^2 +3} - (n+1)) converge or diverge? If it converges, give the limit. If it diverges, give a short explanation.

Sample FINAL exam (under construction. Answers will also be posted)

• 1). Prove by induction d^n/dx^n (sin x) = sin(x + (n{\pi} /2)), for all positive integers n.
• 2).For the series \sum_{n=0}^{\infty} (-1)^n (n+1)^2 2^n x^n determine the interval of convergence.
• Check the endpoints of this interval to determine if the series converges or diverges there.
• 3) Check the following series for convergence or divergence and give a short mathematical reason for your answer
• a) \sum_{n=1}^{\infty} 1/(n + (1/n))
• b) \sum_{n=1}^{\infty} (-1)^n (n+1)/2n
• 4). Using the \epsilon - N definition of limit, prove that lim_{n goes to \infty} (n^2 + n + 1)/ (2n^2 + 3) = 1/2
• 5.If \sum_{n=1}^{\infty} a_n is an absolutely convergent series and \sum_{n=1}^{\infty} b_n is a divergent series, prove that \sum_{n=1}^{\infty} (b_n + (a_n/n)) is a divergent series.
• 6. Check the following series for convergence or divergence and give a short mathematical reason for yor answer:
• a). \sum_{n=2}^{\infty} 1/ n ln(n)
• b). \sum_{n=1}^{\infty} (-1)^n/n^{1/2}
• 7) Use the fact that e^x is an analytic function on the real line. Give an estimate for e^{-1} to within .01
• 8) For the set A = {(-1)^n/(1+(1/n)) | n a natural number}, detemine which of the following exist and, if they do exist, find their value: glb(A), lub(A), max(A), min(A).
  • Possible answers are on the Syllabus page.