Sequences and Their Limits · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.
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Where sequences settle down

Mathematics · Calculus & Analysis · ages 19-20
Name ______________________   Date ____________

  1. For positive integers n, define a_n = 50 when n < 4 and a_n = -3 when n >= 4. What is the limit of this sequence?

    Answer: ______________

  2. For positive integers n, what is the limit of a_n = (4n + 1)/n^2?

    • 0
    • 4
    • 1
    • 5

  3. For positive integers n, what is the limit of a_n = n/(n + 1) as n grows?

    • 1
    • 2
    • 0

  4. To decide whether an infinite series converges, we study its running totals, called partial sums.

    Circle one:   True   False

  5. For positive integers n, let a_n = 2 + 3(-1)^n. This sequence is bounded. Which statement describes its convergence?

    • It converges to 2 because averaging each pair of consecutive terms gives 2.
    • It converges to 5 because every even-indexed term has the value 5.
    • It converges to -1 because every odd-indexed term has the value -1.
    • It diverges because the even and odd terms stay at different values.

  6. Let a_1 = 1 and a_(n+1) = (a_n + 4)/3. The first term is at most 2, and a_n <= 2 implies a_(n+1) <= 2. What does induction establish?

    • Every term is at least 2.
    • Every term is greater than 2.
    • Every term is at most 2.

  7. A real sequence is known to converge to L. Its limit satisfies L^2 = 2 + L and L > 0. Find L.

    Answer: ______________

  8. An infinite series has nth partial sum S_n = 5 - 1/(n + 2), for n >= 1. What is the sum of the series?

    Answer: ______________

  9. Every increasing real sequence that is bounded above converges to a finite limit. Increasing allows consecutive terms to be equal.

    Circle one:   True   False

  10. Let b_1 = 1 and b_(n+1) = (b_n + 6)/3. It has been proved that 0 <= b_n <= 3 for every n. Which valid calculation establishes that the sequence is increasing?

    • b_(n+1) - b_n = (6 - b_n)/3 >= 0
    • b_(n+1) - b_n = 2(3 - b_n)/3 >= 0
    • b_(n+1) - b_n = 2(b_n - 3)/3 >= 0
    • b_(n+1) - b_n = (b_n + 6)/3 >= 0
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Answer key

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Where sequences settle down W1-mt_PQl3Q6n5dc-s1

  1. -3 · The first three terms are 50, but every term from index 4 onward equals -3. The constant tail gives limit -3.
  2. 0 · The expression equals 4/n + 1/n^2. Both terms tend to 0, so the limit is 0.
  3. 1 · Dividing numerator and denominator by n gives 1/(1 + 1/n). Since 1/n tends to 0, the limit is 1.
  4. True · A series converges exactly when its sequence of partial sums approaches a finite number.
  5. It diverges because the even and odd terms stay at different values. · Odd-indexed terms equal 2 - 3 = -1 and even-indexed terms equal 2 + 3 = 5. The sequence cannot approach one number.
  6. Every term is at most 2. · The bound holds for a_1 and is preserved by the rule. Induction gives a_n <= 2 for every positive integer n.
  7. 2 · The equation becomes (L - 2)(L + 1) = 0. Its roots are 2 and -1; L > 0 selects 2.
  8. 5 · The series sum is the limit of S_n. Since 1/(n + 2) tends to 0, S_n tends to 5.
  9. True · This is the monotone convergence theorem. The least upper bound of the terms is their limit.
  10. b_(n+1) - b_n = 2(3 - b_n)/3 >= 0 · Subtracting b_n gives (6 - 2b_n)/3 = 2(3 - b_n)/3. The bound b_n <= 3 makes this nonnegative.
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