Convergence of Sequences, Rigorously · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

Proving a sequence settles down

Mathematics · Calculus & Analysis · ages 20-21
Name ______________________   Date ____________

  1. For a_n equal to 3 over n, what is the smallest integer N such that every n greater than or equal to N gives a_n below 0.04? Give N.

    Answer: ______________

  2. For a_n equal to 1 over n, what is the smallest integer N such that every n greater than or equal to N gives a_n below 0.01? Give N.

    Answer: ______________

  3. Kai says one convergent sequence can have two different limits, 2 and 3. Is Kai right?

    Circle one:   True   False

  4. Suppose a_n tends to 2 and b_n tends to 5. Which step belongs in a proof that a_n plus b_n tends to 7?

    • Split epsilon into two halves and take the larger cutoff.
    • Check one fixed epsilon only, then reuse that answer for every smaller tolerance.
    • Add the two limits first, without choosing cutoffs or bounding an error.

  5. If a_n tends to negative 2 and b_n tends to 3, what does the product law give for a_n b_n?

    • negative 6
    • negative 2 plus 3, which is 1
    • 6
    • The product diverges because one limit is negative

  6. What is the limit of a_n equal to (3n plus 1) over (n plus 2)?

    • 1 over 2
    • It diverges
    • 3

  7. What is the limit of a_n equal to 1 over 2 to the n?

    • 1
    • 0
    • 1 over 2

  8. If a_n tends to 4 and b_n tends to negative 1, what is the limit of a_n plus b_n?

    • 3
    • 5
    • negative 3
    • 0

  9. For a_n equal to 1 over n, what is the smallest integer N such that every n greater than or equal to N gives a_n below 0.002? Give N.

    Answer: ______________

  10. If two sequences converge, taking the larger of their two epsilon cutoffs makes both error bounds hold at once. True or false?

    Circle one:   True   False

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Answer key

For grown-ups. Fold this page away before handing over the rest.

Proving a sequence settles down W1-mt_yB5M-DCH8V-s1

  1. 76 · Three over n below 0.04 means n is greater than 75, so 76 is the first valid cutoff.
  2. 101 · The strict inequality 1 over n below 0.01 means n is greater than 100, so the first valid integer cutoff is 101.
  3. False · Limits are unique: any two candidates must coincide.
  4. Split epsilon into two halves and take the larger cutoff. · Split epsilon into two halves. The two convergence cutoffs then make the triangle inequality bound the sum error below epsilon.
  5. negative 6 · The product law multiplies the limits, so the limit is negative 6.
  6. 3 · Divide top and bottom by n and read the leading coefficients.
  7. 0 · The terms halve each step and stay within any epsilon past some N.
  8. 3 · The sum law adds the two limits, giving 4 plus negative 1 equal to 3.
  9. 501 · The strict inequality 1 over n below 0.002 means n is greater than 500, so the first valid integer cutoff is 501.
  10. True · True. Every later index is past both cutoffs, so both bounds hold and the triangle inequality can combine them.
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