Step by step
The question
Question: find the limit of the sequence a n = (−1)ⁿ⁺¹ over n², as n grows.
an = (−1)n+1n2
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 1.
So the size goes to 0.
as n → ∞ = 1n2 → 0
The sign keeps changing, but the size goes to 0.
Plus or minus a tiny number is still tiny.
So the terms go to 0.
size → 0, so an → 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.
Answer
= converges to 0