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

Where sequences settle down

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

  1. What is the limit of the sequence a_n = (5n minus 1) over (2n + 3)? Give your answer as a decimal.

    Answer: ______________

  2. What is the limit of the sequence a_n = 2n over n + 1?

    • 2
    • 1
    • 1/2
    • 0

  3. What is the limit of the sequence n over n plus 1?

    • 1
    • 2
    • 0

  4. A series adds the terms of a sequence through running totals called partial sums.

    Circle one:   True   False

  5. What is the limit of a_n = (minus 1) to the n over n?

    • 0
    • 1
    • minus 1
    • It diverges by oscillation

  6. A sequence starts at 1 with each next term equal to the current term plus 4, over 3. If a term at most 2 forces the next at most 2, what follows?

    • The sequence exceeds 2 by term 5
    • The bound fails since the rule is curved
    • The sequence is bounded above by 2

  7. A climbing sequence with a ceiling has limit L with L squared equals 2 plus L, L positive. Type L.

    Answer: ______________

  8. An increasing sequence bounded above satisfies L squared = 2 + L at its limit L, with L positive. What is L?

    Answer: ______________

  9. Every climbing sequence with a ceiling settles on some limit.

    Circle one:   True   False

  10. A sequence starts at a_1 = 1 with a_{n+1} = (a_n + 4) over 3. If a_n at most 2 forces a_{n+1} at most 2, what follows?

    • The sequence is bounded above by 2
    • The sequence exceeds 2 by term 5
    • The sequence decreases to 0
    • The bound fails because the map is nonlinear
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Answer key

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Where sequences settle down W1-mt_PQl3Q6n5dc-s1

  1. 2.5 · Leading coefficients dominate: 5 over 2, or 2.5.
  2. 2 · Dividing top and bottom by n gives 2 over 1 plus 1 over n, which tends to 2.
  3. 1 · Divide top and bottom by n: 1 over 1 plus 1 over n tends to 1.
  4. True · Listing terms and adding them ask different questions.
  5. 0 · The magnitude 1 over n squeezes to 0, and the alternating sign cannot stop the squeeze.
  6. The sequence is bounded above by 2 · The base case holds and the bound passes from each term to the next.
  7. 2 · The factored form gives roots 2 and minus 1, so keep 2.
  8. 2 · L squared minus L minus 2 factors as (L minus 2)(L plus 1), and the positive root is 2.
  9. True · That is the monotone bounded promise doing its work.
  10. The sequence is bounded above by 2 · The base case holds since 1 is below 2, and the preserved bound propagates by induction.
Worksheet · LightMySky