The question
Question: find the limit of the sequence a n = 5/n², as n grows.
an = 5n2
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
As n grows, 1/n gets as small as we like.
1n → 0
The bottom grows without end, and the top stays 5.
So the size goes to 0.
as n → ∞ = 5n2 → 0
3 Decide
The terms settle on one number: the sequence converges to 0.
= converges to 0
Careful!
Converges means the terms settle on one number.
Careful: a limit is one number the terms settle on
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.
▶ Watch it solved
The question
Question: find the limit of the sequence a n = (2n² − 1)/(n² + 3), as n grows.
an = 2n2 − 1n2 + 3
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 2.
They get close to 2
2 Divide by the highest power
Divide every term, top and bottom, by n to the 2, the highest power at the bottom.
The value does not change.
an = 2 − 1n21 + 3n2
As n grows, 1/n gets as small as we like: it goes to 0.
So does any number over a power of n.
1n → 0
3 The limit
Every part with n at the bottom goes to 0.
as n → ∞ = 2 − 01 + 0
What is left: 2.
= 2
4 Decide
The terms settle on one number: the sequence converges to 2.
= converges to 2
Careful!
Top and bottom both grow, but that does not make the limit 1.
Divide by the highest power and see what is left.
Careful: not ∞∞ = 1 ✗
5 Check
Let's check with a calculator: term number 1000 is about 2.
Close to 2.
Right!
n = 1000: an ≈ 2
Let's look back.
First: the first terms.
Then: divide by the highest power.
Then: the limit.
Then: decide.
Last: check.
▶ Watch it solved
The question
Question: find the limit of the sequence a n = (−1)ⁿn/(n + 1), as n grows.
an = (−1)nnn + 1
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 jump between plus and minus.
They jump between plus and minus
2 Divide by the highest power
First the size, without the sign.
Divide every term, top and bottom, by n, the highest power at the bottom.
The value does not change.
|an| = 11 + 1n
As n grows, 1/n gets as small as we like: it goes to 0.
So does any number over a power of n.
1n → 0
3 The limit
Every part with n at the bottom goes to 0.
as n → ∞ = 11 + 0
What is left: 1.
= 1
The sign keeps changing, and the size does not go to 0.
The terms jump between about 1 and minus that.
No single number to settle on.
(−1)n: −1, 1, −1, 1, ...
4 Decide
There is no single number at the end: the sequence diverges.
= diverges
Careful!
Converges means the terms settle on one number.
Careful: a limit is one number the terms settle on
5 Check
Let's check with a calculator: term number 1000 is about 0.999, and the next is about −0.999.
Still jumping.
Right!
n = 1000: an ≈ 0.999
Let's look back.
First: the first terms.
Then: divide by the highest power.
Then: the limit.
Then: decide.
Last: check.
▶ Watch it solved
