Step by step
The question
Question: find the limit of the sequence a n = 2ⁿ over 3ⁿ, as n grows.
an = 2n3n
1 The first terms
Work out a few terms: n = 1, 2, 3, then 10 and 100.
Put each n in.
Look at the terms.
They get close to 0.
They get close to 0
2 The limit
Both are powers of n: one power (2/3) to the n.
an = (23)n
The rule for r to the n: a number between −1 and 1 gets smaller each time we multiply by it.
A number bigger than 1 grows.
|r| < 1: rn → 0
r > 1: rn → ∞
2/3 is between 0 and 1: its powers go to 0.
as n → ∞ = (23)n → 0
3 Decide
The terms settle on one number: the sequence converges to 0.
= converges to 0
Careful!
Only a number between minus 1 and 1 gets smaller when we keep multiplying by it.
Careful: rⁿ → 0 only when r is between −1 and 1
4 Check
Let's check with a calculator: term number 1000 is about 0.
Close to 0.
Right!
n = 1000: an ≈ 0
Let's look back.
First: the first terms.
Then: the limit.
Then: decide.
Last: check.
Answer
= converges to 0