Math 312: Real Analysis Final Exam Study Guide | Exams Calculus

Math 312: Real Analysis Fall 2008

Penn State University Section 001

Final Exam Study Guide

The final exam is scheduled for Monday, December 15, from 8:00am to 9:50am in 102 Chem. The exam will

cover material from Chapters 1 through 17 from our textbook. Material from Chapter 22 will be covered during

the last week of classes, but will not appear on the final exam. All questions on the final exam will come from

this study guide. While solutions will not be posted, I will be more than happy to discuss possible solutions to

these problems during office hours.

Students will be expected to know the following definitions as they relate to sequences and/or functions.

•sequence, subsequence

•increasing, decreasing

•monotonic

•upper/lower bound

•bounded

•locally increasing

•locally bounded

•locally positive

•the limit of a sequence

•the limit of a function

•nested intervals

•cluster point

•Cauchy sequence

•maximum/minimum

•supremum/infimum

•sequence of partial sums

•Panconverges

•absolutely convergent

•conditionally convergent

•radius of convergence

•even/odd

•periodic

•continuity

•intermediate value property

•sequentially compact

•derivative

•local max/min

•linearization

•Taylor polynomial

•Taylor series

Students will also be expected to answer (with proper justification) any of the following questions. These

problems are in no particular order, so feel free to use any technique we have studied to answer these questions,

unless otherwise instructed.

1. For each of the following statements, determine if it is true or false.

(a) For all real numbers aand b,|a−b||a+b| ≤ (|a|+|b|)2.

(b) For all nonzero real numbers aand b, if a < b then 1

a>1

(d) The sequence {cos(n)}has a convergent subsequence.

(e) If {an}is bounded, then {an}is convergent.

(f) If {an}converges, then {an}is bounded.

(g) If {an}converges, then {an}is monotonic.

(h) If an∈Sfor all n≥0 and {an}converges, then lim an∈S.

(i) If Panconverges then P|an|converges.

(j) If limn→∞ an= 0 then Panconverges.

2. Consider the sequence {an}for n≥0 where an=1 + 1

201 + 1

211 + 1

22···1 + 1

2n. Show that {an}

converges. Hint: Consider ln anand use the fact that ln(1 + x)< x for x > 0.

3. For what values of xdoes the sequence {xn/n!}converge? What does it converge to?

4. Suppose that {an}is increasing and bounded above. Show that {an}converges to sup{an}.

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Download Math 312: Real Analysis Final Exam Study Guide and more Exams Calculus in PDF only on Docsity!

Math 312: Real Analysis Fall 2008

Penn State University Section 001

Final Exam Study Guide

The final exam is scheduled for Monday, December 15, from 8:00am to 9:50am in 102 Chem. The exam will cover material from Chapters 1 through 17 from our textbook. Material from Chapter 22 will be covered during the last week of classes, but will not appear on the final exam. All questions on the final exam will come from this study guide. While solutions will not be posted, I will be more than happy to discuss possible solutions to these problems during office hours. Students will be expected to know the following definitions as they relate to sequences and/or functions.

sequence, subsequence
increasing, decreasing
monotonic
upper/lower bound
bounded
locally increasing
locally bounded
locally positive
the limit of a sequence
the limit of a function
- nested intervals
- cluster point
- Cauchy sequence
- maximum/minimum
- supremum/infimum
- sequence of partial sums
- ∑^ an converges
- absolutely convergent
- conditionally convergent
- radius of convergence
  - even/odd
  - periodic
  - continuity
  - intermediate value property
  - sequentially compact
  - derivative
  - local max/min
  - linearization
  - Taylor polynomial
  - Taylor series Students will also be expected to answer (with proper justification) any of the following questions. These problems are in no particular order, so feel free to use any technique we have studied to answer these questions, unless otherwise instructed.

For each of the following statements, determine if it is true or false. (a) For all real numbers a and b, |a − b||a + b| ≤ (|a| + |b|)^2. (b) For all nonzero real numbers a and b, if a < b then (^1) a > (^1) b. (c) If {an} is monotonic, then {an} has a convergent subsequence. (d) The sequence {cos(n)} has a convergent subsequence. (e) If {an} is bounded, then {an} is convergent. (f) If {an} converges, then {an} is bounded. (g) If {an} converges, then {an} is monotonic. (h) If an ∈ S for all n ≥ 0 and {an} converges, then lim an ∈ S. (i) If ∑^ an converges then ∑^ |an| converges. (j) If limn→∞ an = 0 then ∑^ an converges.
Consider the sequence {an} for n ≥ 0 where an = (1 + 210 ) (1 + 211 ) (1 + 212 )^ · · · (1 + (^21) n^ ). Show that {an} converges. Hint: Consider ln an and use the fact that ln(1 + x) < x for x > 0.
For what values of x does the sequence {xn/n!} converge? What does it converge to?
Suppose that {an} is increasing and bounded above. Show that {an} converges to sup{an}.

Using only the definition of a limit, show that if lim n→∞ an = L, then lim n→∞ a^2 n = L^2.
Show that if lim n→∞ an = 0 and {bn} is bounded then lim n→∞ anbn = 0.
Suppose that lim n→∞ a 2 n = L and lim n→∞ a 2 n+1 = L. Show that lim n→∞ an = L.
Suppose that lim n→∞ a^2 n = 4. Does {an} converge? Does {|an|} converge?
Determine the value of each of the following limits, if they exist. (a) (^) nlim→∞^ sin( nn)^ (b)^ nlim→∞ sin(π

√n) (c) lim n→∞

√sin(1/n)

Consider the sequence {an} defined by a 0 = 5 and an+1 = 1 + 2/an for n ≥ 0. Show that the sequence converges and determine the value of its limit. Hint: Use Problem 3 from Homework #6.
Let a and b be two positive numbers such that a < b. Let a 1 = √ab and b 1 = (a + b)/2. Define an+1 = √anbn and bn+1 = (an + bn)/2. Show that lim n→∞ an = lim n→∞ bn. Hint: First, show that if an < bn then an < an+1 < bn+1 < bn. Second, show that the nested interval theorem applies.
Prove the Comparison test. In other words, show that if 0 ≤ an ≤ bn for all n ≥ 0 and ∑^ bn converges then ∑^ an converges.
Suppose that

∑^ ∞

an converges to L. Show that

∑^ ∞

(a 2 n + a 2 n+1) also converges to L.

Find the interval of convergence for each of the following power series:

(a)

∑^ ∞

n^3 xn^ (b)

∑^ ∞

3 n n^3 x

3 n (^) (c) ∑^ ∞ n=

(3 + (−1)n)nxn

Show that if f (x) is even and decreasing on R then f (x) is constant.
Show that f (x) = sin(x) is not locally increasing at π/2.
Using the limit definition of continuity, show that f (x) = √x is continuous on [0, ∞).
State and prove the intermediate value theorem.
Show that f (x) = xe−x^ − 1 /π has at least two positive solutions.
Let f (x) be a polynomial of odd degree. Show that f (x) = a has at least one solution for every a ∈ R.
Suppose that f (x) is continuous and positive on [0, ∞) and lim x→∞ f (x) = ∞. Show that g(x) = f (x)/(1 + f (x)) has a minimum value on [0, ∞).
Show that if f (x) is differentiable at a then f (x) is continuous at a.
Show that f (x) =

x^2 if x is rational 0 if x is irrational is differentiable at 0 and not differentiable everywhere else.

Math 312: Real Analysis Final Exam Study Guide, Exams of Calculus

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