courses › Beyond College

Precise facts about limits, bounds, and convergence.

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skills · practice stats

From rounds of this course only: box, review and trial answers are left out. Once a skill has 40 tries, it compares your first 20 tries with your last 20.

• Supremum & infimum par 20 s not tried yet

  worked example

  What is the supremum (least upper bound) of the set {n/(n + 4) : n = 1, 2, 3, …}? Give an exact number or fraction.

  Answer: 1

  1. The terms 1/5, 1/3, 3/7, 1/2, … increase toward 1 without reaching it.
  2. Every term is below 1, and nothing smaller bounds them all, so the supremum is 1.

• Limits that give e par 25 s not tried yet

  worked example

  The limit of (1 + 5/n)²ⁿ as n → ∞ is eᵏ. What is k? Give an exact number or fraction.

  Answer: 10

  1. (1 + a/n)ⁿ → eᵃ, so (1 + 5/n)ⁿ → e⁵.
  2. Raising to the power 2n: (1 + 5/n)²ⁿ = ((1 + 5/n)ⁿ)² → (e⁵)² = e¹⁰.
  3. So k = 10.

• Which is true? coming soon not tried yet

rest ladder

Win 3 of your last 4 rounds and the course rests. A win is 90% right, within 2× the round's par. Pass the review when it comes back and the next rest is longer.

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