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. Suppose a_n tends to 2 and b_n tends to 5. What is the limit of a_n plus b_n?

    • 7
    • 10
    • 3

  2. Suppose a_n tends to 2 and b_n tends to 5. What is the limit of a_n + b_n?

    • 7
    • 10
    • 3
    • 0

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

    Circle one:   True   False

  4. For a_n equal to 1 over n, how large must N be so that n past N forces a_n below 0.01? Give N.

    Answer: ______________

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

    • 1 over 2
    • It diverges
    • 3

  6. The alternating sequence -1, 1, -1, 1, and so on converges to 0.

    Circle one:   True   False

  7. What is the limit of a_n = (3n + 1)/(n + 2)?

    • 3
    • 1/2
    • 0
    • It diverges

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

    • 1
    • 0
    • 1 over 2

  9. For a_n equal to 1 over n, how large must N be so that n past N forces a_n below 0.002? Give N.

    Answer: ______________

  10. A student proves 1 over n tends to 0 by checking epsilon equal to 0.01 only. What is missing?

    • Nothing, one epsilon settles it
    • The argument must work for every positive epsilon, not just one
    • N must be smaller than epsilon
LightMySky · lightmysky.comW1-mt_yB5M-DCH8V-s1

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. 7 · The limit of a sum is the sum of the limits.
  2. 7 · The sum law adds the limits: 2 plus 5 is 7.
  3. False · Kai is wrong. Limits are unique: a convergent sequence has exactly one limit.
  4. 100 · You need n above 100, since 1 over 100 equals 0.01.
  5. 3 · Divide top and bottom by n and read the leading coefficients.
  6. False · Its terms never stay near 0; they keep jumping between clusters.
  7. 3 · Divide top and bottom by n: (3 + 1/n)/(1 + 2/n) tends to 3/1, which is 3.
  8. 0 · The terms halve each step and stay within any epsilon past some N.
  9. 500 · Work backward from the inequality, as in the lesson example.
  10. The argument must work for every positive epsilon, not just one · The definition demands an N for each epsilon, however small.
Worksheet · LightMySky