courses › Beyond College

Real analysis

level 57 course

Precise facts about limits, bounds, and convergence.

Pen and paper is fine · no calculator needed why?

Opens at level 57.

Builds on: Series (not open yet)

the lesson

The idea, the techniques and a tip for each skill, right here. The Learn page adds worked examples for every skill and untimed practice.

Read the lesson · about 3 minutes

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

The supremum or infimum of a set given by a formula in n.

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

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

A limit of 1 plus something over n, raised to a power that grows with n.

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

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

Tips by skill

TipSupremum & infimum: Write out the first few terms and see which way they move. A supremum or infimum can be a limit the terms never reach.
TipLimits that give e: (1 + x/n)ⁿ tends to eˣ. Rewrite the inside as x over n, then multiply x by any number in the power.

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.

skills · practice stats

From rounds of this course only: box, review and test-out 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.

1 day
3 days
7 days
14 days
30 days
60 days
mastered · every 90 days

your rounds

No rounds yet.