Mathematics 242 – Principles of Analysis Practice Final Examination

I. Let A = {x^{2} − 2 : −1 < x < 3} and B = {x : |x − 1| < 4}. Find sup(A ∪ B) and inf(A ∩ B).

II.

A) State the ϵ, N definition of convergence for a sequence of real numbers. B) Identify lim_{n→∞} \frac{5n+1}{n+4}. C) Show that your result in part B is correct using the definition.

III.

A) Show that if x_{n} is a monotone increasing sequence of real numbers that is bounded

above, then x_{n} converges to some real number. B) What general theorem could you use to deduce that the sequence x_{n} = (−1)^{n}+ \frac{(−1)^{n+1}}{n^{2}}

has convergent subsequences? C) Let x_{n} = (−1)^{n} + \frac{(−1)^{n+1}}{n^{2}} . Find a monotone increasing subsequence of (x_{n}) and

identify the limit.

IV. A) Show that if f(x) is continuous at x = c, and x_{n} is any sequence satisfying lim_{n→∞} x_{n} =

c, then lim_{n→∞} f(x_{n}) → f(c). B) Let

f(x) = \{ \begin{matrix}x & if x is a rational number \\ −x & if x is an irrational number \end{matrix}

Is f continuous at x = 1? Why or why not? What about at x = 0?

V. A) State and prove the Intermediate Value Theorem. B) Show that for every real y_{0}, there exists a solution x of the equation \frac{e^{x}−e^{−x}}{2} = y_{0}

VI. (15) Using the definition, compute f′(x) for f(x) = \frac{1}{(x+3)^{2}} .

VII. A) (20) In this part you may use the summation formulas:

∑_{i=1}^{n} 1 = n ∑_{i=1}^{n} i = \frac{n^{2} + n}{2} ∑_{i=1}^{n} i^{2} = \frac{2n^{3} + 3n^{2} + n}{6}.

Show that f(x) = x^{2} + 5 is integrable on [a, b] = [0, 1] by considering upper and lower sums for f and determine the value of ∫_{0}^{1} x^{2} + 5 dx.

B) (15) State and prove the Fundamental Theorem of Calculus, “part 1.”

A) State the definition of convergence for an infinite series ∑_{n=1}^{∞} a_{n}. B) Suppose that in an infinite series ∑_{n=1}^{∞} a_{n}, a_{n} < 0 for all n. Show that if the partial

sum s_{N} satisfies s_{N} > B for all N, then ∑_{n=1}^{∞} a_{n} converges. (Hint: What kind of sequence is {s_{N}}?)

C) Does the series ∑_{n=1}^{∞} \frac{5·3^{n}}{7^{n}} converge? If so, what is the sum of the series? D) Use the Integral Test to determine whether

∑_{n=2}^{∞} \frac{1}{n(ln n)^{2}}

converges. E) Use the Comparison Test to determine whether the series

∑_{n=1}^{∞} \frac{n^{2}}{n^{3.9} + 1}

converges absolutely.

IX. A) Noting that

lim_{h→0} \frac{ln(1 + h)}{h} = lim_{h→0} \frac{ln(1 + h) − ln(1)}{h},

evaluate the limit. (You may use any calculus facts you need here without proof.) B) Let h = \frac{1}{n} in the limit from part A, and use that result and the theorem from question

IV A on this exam with f(x) = e^{x} to compute

lim_{n→∞} (1 + \frac{1}{n})^{n} .