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PEI Math 521K practice questions

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Question 1Real Number System and Elementary Logic

Which statement is the correct negation of the proposition "Every even integer greater than 2 is a sum of two primes"?

  1. There exists an even integer greater than 2 that is not a sum of two primes. Correct answer.
  2. Every even integer greater than 2 is not a sum of two primes.
  3. No even integer greater than 2 is a sum of two primes.
  4. There exists an even integer greater than 2 that is a sum of two primes.

Explanation

Negating a universal statement yields an existential statement with the predicate negated: the negation of (for all x, P(x)) is (there exists x such that not P(x)). Here P(x) is 'x is a sum of two primes' restricted to even integers above 2, so the negation asserts one such integer fails to be a sum of two primes.

Question 2Real Number System and Elementary Logic

For real numbers a and b, the statement "a·b = 0 implies a = 0 or b = 0" is best described as which property of the real number system?

  1. The no-zero-divisors property of the reals Correct answer.
  2. The Archimedean property
  3. The density of the rationals
  4. The completeness axiom

Explanation

In a field, if a product equals zero then at least one factor is zero; this is the statement that the reals have no zero divisors. Completeness concerns least upper bounds, density concerns rationals between reals, and the Archimedean property concerns multiples exceeding any real.

Question 3Real Number System and Elementary Logic

Consider the implication "If n is divisible by 6, then n is divisible by 3." Which statement is its contrapositive?

  1. If n is not divisible by 3, then n is not divisible by 6. Correct answer.
  2. If n is divisible by 3, then n is divisible by 6.
  3. If n is divisible by 6, then n is not divisible by 3.
  4. If n is not divisible by 6, then n is not divisible by 3.

Explanation

The contrapositive of P implies Q is (not Q) implies (not P), and it is logically equivalent to the original. Swapping and negating both parts gives: if n is not divisible by 3, then n is not divisible by 6.

Question 4Real Number System and Elementary Logic

Which of the following is a true statement about the set of rational numbers Q as a subset of the reals?

  1. Q contains a least upper bound for every nonempty subset bounded above.
  2. Q is neither open nor closed as a subset of R. Correct answer.
  3. Q is closed under taking square roots of its nonnegative elements.
  4. Q is finite in every bounded interval.

Explanation

The rationals are dense but do not contain all their limit points, and their complement (the irrationals) is also dense, so Q is neither open nor closed in R. It fails closure under square roots and lacks the least-upper-bound property, and every interval contains infinitely many rationals.

Question 5Real Number System and Elementary Logic

Let A and B be subsets of R. Which identity correctly describes the complement of A ∩ B in R?

  1. R \ (A ∪ B)
  2. (R \ A) ∪ (R \ B) Correct answer.
  3. (R \ A) ∩ (R \ B)
  4. A ∪ (R \ B)

Explanation

De Morgan's law states the complement of an intersection is the union of the complements: R \(A ∩ B) = (R \ A) ∪ (R \ B). An element fails to be in A ∩ B precisely when it misses A or misses B.

Question 6Real Number System and Elementary Logic

Which statement correctly expresses the density of the rationals in the reals?

  1. Between any two distinct rationals there is an integer.
  2. Between any two distinct real numbers there are infinitely many rational numbers. Correct answer.
  3. Every rational number lies between two integers.
  4. Every real number is rational.

Explanation

Density means that for any reals a < b there exists a rational q with a < q < b; iterating this yields infinitely many rationals in the interval. The other options misstate the relationship between rationals, reals, and integers.

Question 7Real Number System and Elementary Logic

The statement "For every real x there exists a natural number n with n > x" is known as which property?

  1. The Archimedean property Correct answer.
  2. The least-upper-bound property
  3. The well-ordering principle
  4. The Bolzano-Weierstrass property

Explanation

The Archimedean property of the reals says the natural numbers are unbounded above, so each real is exceeded by some natural number. Well-ordering concerns least elements of subsets of naturals, and the other two are about suprema and accumulation points.

Question 8Real Number System and Elementary Logic

Which of these arguments is a valid application of modus tollens?

  1. If x is rational then 2x is rational; x is rational; therefore 2x is rational.
  2. If x is rational then 2x is rational; 2x is not rational; therefore x is not rational. Correct answer.
  3. If x is rational then 2x is rational; x is not rational; therefore 2x is not rational.
  4. If x is rational then 2x is rational; 2x is rational; therefore x is rational.

Explanation

Modus tollens has the form: P implies Q, not Q, therefore not P. Only the first option denies the consequent, which is the valid pattern; the others affirm the consequent or deny the antecedent, both invalid.

Question 9Real Number System and Elementary Logic

Let S be a nonempty subset of R that is bounded above, and let u be an upper bound of S. Which condition makes u the supremum of S?

  1. u belongs to S.
  2. There exists s in S with s = u.
  3. Every element of S is less than u.
  4. For every epsilon > 0 there exists s in S with u - epsilon < s. Correct answer.

Explanation

The supremum is the least upper bound, so no smaller number bounds S. That is exactly the condition that for every positive epsilon some element of S exceeds u minus epsilon. Merely being an upper bound, or being an element of S, does not guarantee leastness.

Question 10Real Number System and Elementary Logic

Which statement is the correct negation of "There exists a real number x such that x squared equals 2 and x is rational"?

  1. There exists a real x such that x squared does not equal 2 or x is irrational.
  2. For every real x, x squared does not equal 2.
  3. There exists a real x such that x squared equals 2 and x is irrational.
  4. For every real x, x squared equals 2 implies x is irrational. Correct answer.

Explanation

Negating an existential gives a universal with the whole conjunction negated: for all x, not (x squared equals 2 and x rational), which by De Morgan is: x squared equals 2 implies x irrational. This is the standard statement that the square root of 2 is irrational.

Question 11Real Number System and Elementary Logic

Which pair of statements about a real number x is logically equivalent?

  1. "x > 3" and "x is not less than 3"
  2. "x > 3" and "x is not equal to 3"
  3. "x > 3" and "x is not less than or equal to 3" Correct answer.
  4. "x > 3" and "x is greater than or equal to 3"

Explanation

The negation of x ≤ 3 is x > 3, since the order on the reals is total and trichotomy holds. Saying x is not less than or equal to 3 forces x strictly greater than 3, so the two statements coincide.

Question 12Real Number System and Elementary Logic

Which statement correctly describes the relationship between an implication and its converse?

  1. An implication implies its converse whenever the hypothesis is true.
  2. An implication and its converse are equivalent exactly when the converse is true.
  3. An implication and its converse are always logically equivalent.
  4. An implication and its converse need not be equivalent, and neither implies the other in general. Correct answer.

Explanation

The converse of P implies Q is Q implies P, and these are not equivalent in general; for example, x = 2 implies x squared = 4, but x squared = 4 does not imply x = 2. Only the contrapositive is always equivalent to the original.

Question 13Real Number System and Elementary Logic

Which set of real numbers is equal to the intersection over all positive integers n of the open intervals (0, 1/n)?

  1. The set of all positive reals
  2. The interval (0, 1)
  3. The empty set Correct answer.
  4. The set containing only 0

Explanation

No positive real x lies in every interval (0, 1/n), because by the Archimedean property some n satisfies 1/n < x. Zero is excluded since the intervals are open at 0, so the intersection is empty.

Question 14Real Number System and Elementary Logic

Suppose P and Q are propositions and the compound statement "P or Q" is false. What follows about P and Q?

  1. P is true and Q is false.
  2. At least one of P and Q is false.
  3. Both P and Q are true.
  4. Both P and Q are false. Correct answer.

Explanation

A disjunction is false only when both disjuncts are false, since a single true disjunct would make the whole statement true. Hence P and Q must each be false.

Question 15Real Number System and Elementary Logic

Which statement is a correct use of the well-ordering principle for the natural numbers?

  1. Every nonempty subset of the reals has a least element.
  2. Every subset of the natural numbers has a greatest element.
  3. Every nonempty subset of the natural numbers has a least element. Correct answer.
  4. Every bounded set of natural numbers has a supremum in the naturals.

Explanation

The well-ordering principle states that every nonempty set of natural numbers contains a least element. It fails for the reals, and it says nothing about greatest elements or suprema.

Question 16Real Number System and Elementary Logic

Let A be a subset of R. Which condition is equivalent to A being closed in R?

  1. A contains a maximum and a minimum.
  2. The complement of A is closed.
  3. A contains every limit point of sequences from A that converge in R. Correct answer.
  4. Every point of A is an interior point of A.

Explanation

A set is closed exactly when it contains all its limit points, equivalently the limit of every convergent sequence in A lies in A. The other options describe bounded extrema, complement closure (which would make A open, not closed), and openness.

Question 17Real Number System and Elementary Logic

Consider the claim: "If a real number x satisfies x squared < 4, then x < 2." Which statement about this claim is accurate?

  1. The claim is false because x could equal 2.
  2. The claim is true, but its converse is false because x could be less than -2. Correct answer.
  3. The claim is false because x could equal -3.
  4. The claim is true, and its converse is also true.

Explanation

If x squared < 4 then x lies between -2 and 2, so certainly x < 2, making the claim true. The converse, x < 2 implies x squared < 4, fails for x = -3, whose square is 9.

Question 18Real Number System and Elementary Logic

Which statement correctly expresses the completeness axiom for the real numbers?

  1. Every nonempty subset of R that is bounded above has a least upper bound in R. Correct answer.
  2. Every nonempty subset of R has a greatest lower bound in R.
  3. Every bounded sequence in R converges.
  4. Every Cauchy sequence in R is bounded.

Explanation

Completeness asserts that any nonempty set of reals bounded above possesses a supremum in R. The unqualified version without a boundedness hypothesis is false, and the sequence statements are consequences or weaker facts, not the axiom itself.

Question 19Real Number System and Elementary Logic

Which of the following statements about the real numbers is false?

  1. Between any two distinct reals there is an irrational number.
  2. Every real number is either rational or irrational.
  3. The product of a nonzero rational and an irrational is irrational.
  4. The sum of two irrational numbers is always irrational. Correct answer.

Explanation

The sum of two irrationals can be rational: the square root of 2 plus its negative equals 0. The other three statements are true by definition of irrationals, by a standard closure argument, and by density of the irrationals.

Question 20Real Number System and Elementary Logic

Which logical form corresponds to a proof by contradiction of the statement "there is no rational number whose square is 2"?

  1. Assume some irrational q satisfies q squared equals 2 and derive a contradiction.
  2. Prove directly that every rational q has q squared not equal to 2 without any assumption.
  3. Assume every rational q satisfies q squared not equal to 2 and derive a contradiction.
  4. Assume some rational q satisfies q squared equals 2 and derive a contradiction. Correct answer.

Explanation

A proof by contradiction negates the goal and shows the negation is impossible. The goal is that no rational squares to 2, so we assume a rational q with q squared equals 2 and derive a contradiction.

Question 21Sequences, Series, and Limits

What is the value of the limit of the sequence a_n = (3n² + 2n)/(n² + 5) as n tends to infinity?

  1. 1
  2. 3 Correct answer.
  3. 2
  4. 0

Explanation

Dividing numerator and denominator by n² gives (3 + 2/n)/(1 + 5/n²). As n grows without bound, 2/n and 5/n² tend to 0, so the quotient tends to 3/1 = 3. The limit of a rational sequence equals the ratio of leading coefficients when degrees match.

Question 22Sequences, Series, and Limits

A sequence is defined recursively by a_1 = 2 and a_{n+1} = (a_n + 6)/2. Assuming the sequence converges to a limit L, what is L?

  1. 3
  2. 6 Correct answer.
  3. 4
  4. 2

Explanation

If a_n tends to L, then both a_{n+1} and a_n tend to L, so L = (L + 6)/2. Solving gives 2L = L + 6, hence L = 6. The fixed point of the recursion is the only possible limit.

Question 23Sequences, Series, and Limits

What is the sum of the geometric series 1 + 1/3 + 1/9 + 1/27 + ... ?

  1. 3/2 Correct answer.
  2. 1
  3. 2/3
  4. 9/4

Explanation

A geometric series with first term a = 1 and ratio r = 1/3 converges because |r| < 1. Its sum is a/(1 - r) = 1/(1 - 1/3) = 1/(2/3) = 3/2.

Question 24Sequences, Series, and Limits

The series with positive terms sum of b_n converges. What can be concluded about the series sum of b_n²?

  1. It may diverge, since squaring can enlarge small terms.
  2. It converges. Correct answer.
  3. It diverges to infinity.
  4. Nothing can be concluded without more information.

Explanation

Since sum b_n converges, its terms satisfy b_n → 0, so eventually 0 < b_n < 1. For such terms b_n² < b_n, and the comparison test forces sum b_n² to converge.

Question 25Sequences, Series, and Limits

What is the limit of the sequence a_n = n·sin(1/n) as n tends to infinity?

  1. 0
  2. 1 Correct answer.
  3. Infinity
  4. 1/2

Explanation

Write a_n = sin(1/n)/(1/n) and let x = 1/n, which tends to 0. The standard limit sin(x)/x → 1 as x → 0 gives a_n → 1.

Question 26Sequences, Series, and Limits

Which statement correctly describes the harmonic series sum from n=1 to infinity of 1/n?

  1. It diverges because its terms do not tend to 0.
  2. It converges because its terms tend to 0.
  3. It diverges even though its terms tend to 0. Correct answer.
  4. It converges to 1.

Explanation

The terms 1/n tend to 0, but the partial sums grow without bound, so the series diverges. This shows that the term test is only necessary, not sufficient, for convergence.

Question 27Sequences, Series, and Limits

Apply the ratio test to the series sum of n!/3^n. What is the conclusion?

  1. The series converges to 3.
  2. The series converges because the ratio tends to 1/3.
  3. The terms grow without bound, so the series diverges. Correct answer.
  4. The test is inconclusive.

Explanation

The ratio of consecutive terms is (n+1)!/3^{n+1} divided by n!/3^n, which equals (n+1)/3. This ratio tends to infinity, so the terms do not tend to 0 and the series diverges.

Question 28Sequences, Series, and Limits

Which of the following series converges conditionally?

  1. The series sum of (-1)^n
  2. The series sum of 1/n²
  3. The alternating series sum of (-1)^n/n Correct answer.
  4. The series sum of 1/n

Explanation

The alternating series sum of (-1)^n/n converges by the alternating series test, but the series of absolute values is the harmonic series, which diverges. Convergence without absolute convergence is conditional convergence.

Question 29Sequences, Series, and Limits

What is the limit of the sequence a_n = (2^n)/(3^n + 1) as n tends to infinity?

  1. 1
  2. 0 Correct answer.
  3. Infinity
  4. 2/3

Explanation

Divide numerator and denominator by 3^n to get (2/3)^n divided by (1 + 3^{-n}). Since (2/3)^n tends to 0 and the denominator tends to 1, the limit is 0.

Question 30Sequences, Series, and Limits

The sequence a_n is defined by a_1 = 1 and a_{n+1} = sqrt(2 + a_n). Which statement about this sequence is true?

  1. It decreases and converges to 1.
  2. It increases and is bounded above by 2, so it converges to 2. Correct answer.
  3. It oscillates and does not converge.
  4. It is unbounded and diverges to infinity.

Explanation

By induction each term lies below 2 and the sequence is increasing, since sqrt(2 + a_n) exceeds a_n whenever a_n < 2. A bounded monotone sequence converges, and the limit L satisfies L = sqrt(2 + L), giving L = 2.

Question 31Sequences, Series, and Limits

What is the radius of convergence of the power series sum of x^n/n?

  1. 0
  2. 2
  3. Infinity
  4. 1 Correct answer.

Explanation

The coefficients are 1/n, and the ratio of consecutive coefficients tends to 1, so the radius of convergence is 1. At x = 1 the series is harmonic and diverges, while at x = -1 it converges conditionally.

Question 32Sequences, Series, and Limits

What is the value of the telescoping sum from n=1 to infinity of 1/(n(n+1))?

  1. 1 Correct answer.
  2. 2
  3. 1/2
  4. It diverges.

Explanation

Using partial fractions, 1/(n(n+1)) = 1/n - 1/(n+1). The partial sum collapses to 1 - 1/(N+1), which tends to 1 as N grows.

Question 33Sequences, Series, and Limits

A sequence satisfies 0 ≤ a_n ≤ 1/n² for every n. What follows from the squeeze theorem?

  1. The sequence a_n diverges.
  2. The series sum of a_n equals 1.
  3. The series sum of a_n converges. Correct answer.
  4. The series sum of a_n diverges.

Explanation

Since 0 ≤ a_n ≤ 1/n² and the p-series sum of 1/n² converges, the comparison test gives convergence of sum a_n. The squeeze also shows a_n → 0, consistent with convergence.

Question 34Sequences, Series, and Limits

Which sequence is bounded but divergent?

  1. a_n = 1/n
  2. a_n = 2^n
  3. a_n = (-1)^n Correct answer.
  4. a_n = n

Explanation

The sequence (-1)^n alternates between -1 and 1, so it stays within a fixed range but has no single limit. Boundedness does not by itself guarantee convergence.

Question 35Sequences, Series, and Limits

What is the sum of the series sum from n=0 to infinity of (1/2)^n?

  1. 3/2
  2. 1
  3. It diverges.
  4. 2 Correct answer.

Explanation

This geometric series has first term 1 and ratio 1/2, so its sum is 1/(1 - 1/2) = 2. Convergence is guaranteed because the ratio has absolute value below 1.

Question 36Sequences, Series, and Limits

The series sum of a_n converges. Which statement must be true?

  1. The series sum of a_n² converges.
  2. The series sum of |a_n| converges.
  3. The sequence a_n tends to 0. Correct answer.
  4. The sequence a_n is monotone decreasing.

Explanation

Convergence of a series forces its terms to tend to 0, since each term is the difference of consecutive partial sums. The converse fails, as the harmonic series shows.

Question 37Sequences, Series, and Limits

What is the limit of the sequence a_n = (1 + 1/n)^n as n tends to infinity?

  1. 1
  2. Infinity
  3. 2
  4. e Correct answer.

Explanation

This is the standard limit defining the base of the natural logarithm, tending to e ≈ 2.71828. The sequence increases and is bounded above, so the limit exists.

Question 38Sequences, Series, and Limits

Apply the integral test to the p-series sum of 1/n^p with p > 0. What is the conclusion?

  1. It converges exactly when p < 1.
  2. It converges exactly when p > 1. Correct answer.
  3. It converges for every p > 0.
  4. It diverges for every p > 0.

Explanation

The function 1/x^p is positive, continuous, and decreasing for x ≥ 1, so the integral test applies. The integral of x^{-p} converges only when p > 1, giving the same conclusion for the series.

Question 39Sequences, Series, and Limits

Which of these series diverges by the nth-term test?

  1. The series sum of 1/2^n
  2. The series sum of n/(n+1) Correct answer.
  3. The series sum of 1/n²
  4. The series sum of (-1)^n/n

Explanation

The terms n/(n+1) tend to 1 rather than 0, so the series must diverge. The nth-term test detects divergence whenever the terms fail to approach zero.

Question 40Sequences, Series, and Limits

The sequence a_n is defined by a_1 = 3 and a_{n+1} = a_n/2. What is the limit of a_n?

  1. 3/2
  2. 0 Correct answer.
  3. 3
  4. It does not converge.

Explanation

The recursion multiplies by 1/2 each step, so a_n = 3·(1/2)^{n-1}. Since the ratio has absolute value below 1, the powers tend to 0 and the sequence converges to 0.

Question 41Sequences, Series, and Limits

Let S_N be the Nth partial sum of a series with nonnegative terms. If the sequence S_N is bounded above, what follows?

  1. The series converges. Correct answer.
  2. The series diverges.
  3. The series converges only if the terms are monotone.
  4. The terms must eventually equal 0.

Explanation

With nonnegative terms the partial sums form an increasing sequence. A monotone increasing sequence that is bounded above converges, and its limit is the sum of the series.

Question 42Sequences, Series, and Limits

What is the interval of convergence of the power series sum of x^n?

  1. The open interval (-1, 1) Correct answer.
  2. All real numbers
  3. The interval (0, 1)
  4. The closed interval [-1, 1]

Explanation

This geometric series converges exactly when |x| < 1, giving the open interval (-1, 1). At both endpoints the terms fail to tend to 0, so the endpoints are excluded.

Question 43Sequences, Series, and Limits

Which statement about the alternating series sum of (-1)^{n+1}/n is correct?

  1. It converges absolutely.
  2. It diverges because the harmonic series diverges.
  3. It converges to ln 2. Correct answer.
  4. It converges to 1.

Explanation

This is the alternating harmonic series, which converges by the alternating series test to ln 2. It converges conditionally, since the series of absolute values is the divergent harmonic series.

Question 44Sequences, Series, and Limits

What is the limit of the sequence a_n = (n + 5)/(2n - 1) as n tends to infinity?

  1. 1/2 Correct answer.
  2. 2
  3. 1
  4. 0

Explanation

Divide numerator and denominator by n to get (1 + 5/n)/(2 - 1/n). As n grows, the small fractions vanish and the limit is 1/2, the ratio of leading coefficients.

Question 45Sequences, Series, and Limits

The root test is applied to the series sum of (n/(2n+1))^n. What is the conclusion?

  1. The series converges because the nth root tends to 0.
  2. The series diverges because the nth root tends to 1.
  3. The series converges because the nth root tends to 1/2. Correct answer.
  4. The test is inconclusive.

Explanation

The nth root of the nth term is n/(2n+1), which tends to 1/2. Since this limit is less than 1, the root test guarantees absolute convergence.

Question 46Sequences, Series, and Limits

Consider the sequence defined by a_1 = 2 and a_{n+1} = 3 + a_n/2 for n ≥ 1. The sequence converges to a limit L. What is L?

  1. 6 Correct answer.
  2. 4
  3. 2
  4. 3

Explanation

For a convergent recursive sequence, the limit satisfies L = 3 + L/2. Solving gives L/2 = 3, so L = 6. The fixed point is the only possible limit, and the contraction factor 1/2 guarantees convergence to it.

Question 47Sequences, Series, and Limits

Let (a_n) be a sequence of real numbers and suppose that the subsequences (a_{2n}) and (a_{2n+1}) both converge to the same real number L. Which conclusion is valid?

  1. The series sum of a_n converges to L
  2. Only the even-indexed subsequence converges
  3. The whole sequence (a_n) converges to L Correct answer.
  4. The sequence (a_n) is bounded but may diverge

Explanation

Every index n is either even or odd, so every term of (a_n) belongs to one of the two subsequences. Given epsilon > 0, choose N so that both subsequences are within epsilon of L beyond their thresholds; then all terms past the larger threshold are within epsilon of L. Hence the full sequence converges to L.

Question 48Sequences, Series, and Limits

A series has partial sums S_N = (3N^2 + 5)/(N^2 + 2). What is the value of the series, and does it converge?

  1. It converges to 3 Correct answer.
  2. It diverges to infinity
  3. It converges to 0
  4. It converges to 5/2

Explanation

A series converges exactly when its sequence of partial sums converges, and its sum equals that limit. Dividing numerator and denominator by N^2 shows S_N tends to 3, so the series converges with sum 3.

Question 49Sequences, Series, and Limits

The terms of a series satisfy a_n > 0 for all n and a_{n+1}/a_n tends to 1/4 as n tends to infinity. What does the ratio test conclude?

  1. The ratio test is inconclusive
  2. The series diverges to infinity
  3. The series converges conditionally but not absolutely
  4. The series converges absolutely Correct answer.

Explanation

The ratio test compares the limiting ratio r = lim |a_{n+1}/a_n| with 1. Here r = 1/4 < 1, so the series converges absolutely. Absolute convergence follows because all terms are already positive.

Question 50Sequences, Series, and Limits

Let (a_n) be a sequence with a_n > 0 for all n, and suppose the series sum of a_n converges. Which statement about the sequence (n a_n) must be true?

  1. It diverges to infinity
  2. It is bounded Correct answer.
  3. It is eventually increasing
  4. It converges to 1

Explanation

If sum a_n converges with positive terms, then the tail sums T_n = a_n + a_{n+1} + ... tend to 0. Since the terms are positive and decreasing along the tail comparison, n a_n ≤ 2(a_{floor(n/2)+1} + ... + a_n) for large n, so n a_n is bounded. Thus the product is bounded even though a_n itself tends to 0.