Real analysis
Precise facts about limits, bounds, and convergence.
Pen and paper is fine · no calculator needed why?
Opens at level 57. You're level 1. You can read and practice here now.
the idea
Real analysis makes calculus exact. Two ideas carry this course.
The supremum of a set is its least upper bound: the smallest number no member is above. The infimum is the greatest lower bound. Neither has to be a member: 1/2, 2/3, 3/4 and so on never reach 1, yet 1 is their supremum.
As n grows, (1 + x/n)ⁿ gets as close as you like to eˣ. With x = 1, that limit is the number e.
techniques
List the first few terms
- Write out the terms for n = 1, 2, 3 and 4.
- If they rise, the first term is the infimum and their limit is the supremum.
- If they fall, the first term is the supremum and their limit is the infimum.
- If the sign flips back and forth, look at the odd and even terms separately.
worked example
What is the infimum (greatest lower bound) of the set {2 + 3/n : n = 1, 2, 3, …}? Give an exact number or fraction.
- The terms are 5, 7/2, 3, 11/4 and so on.
- They fall toward 2 and never reach it.
- Nothing larger than 2 is below them all, so the infimum is 2.
Answer: 2
Match the limit to eˣ
- Use the rule: (1 + x/n)ⁿ tends to eˣ, for any fixed x, positive or negative.
- Write the inside as x/n. For example, 1/(4n) is (1/4)/n, so x is 1/4.
- If the power is bn instead of n, the limit is e to the power bx.
worked example
The limit of (1 − 2/n)³ⁿ as n → ∞ is eᵏ. What is k? Give an exact number or fraction.
- The inside is 1 + (−2)/n, so x is −2.
- The power is 3n, so k is −2 × 3, which is −6.
Answer: −6
watch out for
- Giving the smallest term when asked for the supremum. The supremum is the least upper bound.
- Thinking the supremum must be a member of the set. The terms 1 − 1/n never reach 1, yet 1 is the supremum.
- Saying the limit is 1 because the base tends to 1. The power grows at the same time, and the two pull against each other.
- Forgetting to divide by the number next to n: in (1 + 1/(4n))ⁿ, x is 1/4, not 1.
practice
Supremum & infimum
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
- The terms 1/5, 1/3, 3/7, 1/2, … increase toward 1 without reaching it.
- Every term is below 1, and nothing smaller bounds them all, so the supremum is 1.
Limits that give e
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 + a/n)ⁿ → eᵃ, so (1 + 5/n)ⁿ → e⁵.
- Raising to the power 2n: (1 + 5/n)²ⁿ = ((1 + 5/n)ⁿ)² → (e⁵)² = e¹⁰.
- So k = 10.