Infinite Series and the Geometric Series · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

Adding infinitely many terms

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

  1. What is the sum of the geometric series 1 + 1/2 + 1/4 + 1/8 and so on?

    Answer: ______________

  2. Type the sum of 1 plus 1 over 2 plus 1 over 4 and so on.

    Answer: ______________

  3. A series is defined as the limit of its partial sums.

    Circle one:   True   False

  4. When does a geometric series have a finite sum?

    • The absolute value of the ratio is below 1
    • The ratio is positive
    • The first term is smaller than 1

  5. The harmonic series has terms tending to zero, yet it diverges. What does this show?

    • Terms tending to zero does not guarantee convergence
    • Terms tending to zero guarantees convergence
    • The nth term test is wrong
    • Partial sums are useless for divergence

  6. Type the sum of 4 plus 4 over 3 plus 4 over 9 and so on.

    Answer: ______________

  7. Terms 1 over n times n plus 1 give partial sums N over N plus 1. Type the sum of the series.

    Answer: ______________

  8. The series with terms 1 over n(n + 1) telescopes with partial sums n over n + 1. What is its sum?

    Answer: ______________

  9. Which of these series must diverge?

    • One whose terms shrink to zero and decrease
    • One whose partial sums stay below 10
    • One whose nth term tends to 5

  10. Which of these series must diverge?

    • One whose nth term tends to 5
    • One whose nth term tends to 0 and decreases
    • One whose partial sums stay below 10
    • One whose terms are smaller than 1 over n squared
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Answer key

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

Adding infinitely many terms W1-mt_EPE-ERFFQt-s1

  1. 2 · First term 1 with ratio 1 half gives 1 over 1 minus 1 half, or 2.
  2. 2 · First term 1 with ratio 1 half gives 1 over 1 minus 1 half.
  3. True · Add finitely many terms first, then let the count grow.
  4. The absolute value of the ratio is below 1 · Only a small ratio lets the stray power vanish in the limit.
  5. Terms tending to zero does not guarantee convergence · Vanishing terms are necessary for convergence but not sufficient, and the harmonic series is the classic counterexample.
  6. 6 · First term 4 with ratio 1 third gives 4 over 2 thirds.
  7. 1 · Neighbours cancel and N over N plus 1 tends to 1.
  8. 1 · Partial fractions split each term into 1 over n minus 1 over n + 1, neighbours cancel, and n over n + 1 tends to 1.
  9. One whose nth term tends to 5 · Terms refusing to vanish break the necessary condition for convergence.
  10. One whose nth term tends to 5 · Terms refusing to vanish break the necessary condition for convergence, so divergence is forced.
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