Sample Advanced Calculus 312 Problems - study guide 1. (a) Define the terms upper bound and least upper bound. (b) State the Least Upper Bound Axiom. (c) Suppose that {xn } is an increasing sequence of real numbers, which is bounded above. Prove that the sequence {xn } converges. 2. Give the definition of compactness. Give an example of a non-empty bounded subset of R which is not compact. Find an open cover of your set which does not have a finite subcover. 3. (a) Give the definition of measure zero. (b) Show that the countable union of sets of measure zero also has measure zero. (c) Show that the interval [0, 1] does not have measure zero. Hint: Use a proof by contradiction. Suppose the interval does have measure zero, and apply the definition of measure zero with ε = 1/2. Then use compactness of the interval and derive a contradiction. (d) Conclude that the set of real numbers is uncountable. 4. Suppose f : [a, b] → R is bounded. Define the oscillation Ω(f, I) of f over an interval I, and the oscillation ω(f, x) of f at a point x. Show that f is continuous at a point x if, and only if, ω(f, x) = 0. 5. Suppose f : [a, b] → R is bounded. (a) What is Riemann’s condition for f to be integrable? (b) Give an example of a bounded function f : [0, 1] → R which is not Riemann integrable. Explain. (c) Suppose f : [a, b] → R is continuous. Show that f is integrable. 6. Show that the uniform limit of continuous functions is continuous. 7. Give an example, if possible, of a sequence of continuous functions which converges pointwise to a function which is not continuous. 8. Suppose that {fn (x)} is a sequence of Riemann-integrable functions which converges uniformly on [0, 1] to the limit f (x). Show that f is Riemann-integrable, and Z 1 Z 1 fn (x) dx = f (x) dx. lim n→∞ 0 0 9. Give an example of a sequence {fn (x)} of continuous functions which converges (pointwise) to a limit f (x) on [0, 1], for which Z 1 Z 1 lim fn (x) dx 6= f (x) dx. n→∞ 0 0 1 10. Give an example, if possible, of a sequence {fn } of continuous functions defined on I = [0, 1]R which converges to 0 pointwise, so that the convergence is not uniform, and 1 limn→∞ 0 fn (x) dx = 0. 11. Show that absolute convergence implies convergence for series in a complete normed linear space. PN  1 ikx 12. Sum the geometric series DN (x) = 2π e to show that k=−N  1 1 + 2 cos(x) + 2 cos(2x) + 2 cos(3x) + · · · + 2 cos(N x) 2π sin((N + 1/2)x) = 2π sin(x/2) DN (x) = if x is not a multiple of 2π. P n 13. Consider the power series ∞ n=0 an z where an ∈ C and z ∈ C. Define the “radius of convergence” by R = sup{r ≥ 0 : ∃M so that |an |rn ≤ M ∀n}. Let ρ < R. Show that the series converges absolutely and uniformly on the closed disc {z ∈ C : |z| ≤ ρ}. Hint: Fix r with ρ < r < R. Then for |z| ≤ ρ we have |an z n | ≤ |an |ρn = |an |rn (ρ/r)n with 0 ≤ ρ/r < 1. P n−1 has the same radius of convergence as the 14. Show P that the power series ∞ n=1 an nz n a z . series ∞ n=0 n 15. (a) Suppose A > 0 and Aλ2 + Bλ + C ≥ 0 for all real numbers λ. Show that B 2 − 4AC ≤ 0. (b) Use this result to prove the Cauchy-Schwarz inequality for an inner product. (c) Use the Cauchy-Schwarz inequality to derive the triangle inequality for the norm p defined by ||v|| = (v, v). 16. Use the triangle inequality to show that ||x|| − ||y|| ≤ ||x + y||. 17. State and prove a version of Taylor’s Theorem. 18. Estimate the error term for ex = 1 + x + 2!1 x2 + · · · + n!1 xn + Rn (x) and show that Rn (x) → 0 as n → ∞, for any x. P 1 k Thus ez is given by the power series ∞ k=0 k! z with radius of convergence R = +∞. 19. (a) Find the Taylor series for cos(x) and sin(x) and the radii of convergence. (b) Find the Taylor series for ln(1 + x) and the radius of convergence. (c) Find the Taylor series for arctan(x) and the radius of convergence. Justify all your work. 20. Suppose 0 < α < 1. Show that nαn → 0 as n → ∞. 2 Advanced Calculus 2 – Exam Study Guide 1. Q. Write the question for number 1 exactly as written in the sample question list in bold. A. i) Write out each step of the solution using roman numeral format as demonstrated in this study guide example ii) If possible, please try to keep the solutions short, to the point, and make it easy to memorize iii) However, please ensure that the solutions are 100% correct and fully answers the question iv) Please use Cambria font if possible and 12-point font 2. Q. Write the question for number 2 exactly as written in the sample question list in bold. A. i) Write the solution to the question using the same format as I described in question 1 
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