Step by step
The question
Question: find the limit of the sequence a n = (−1)ⁿ, as n grows.
an = (−1)n
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
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, ...
3 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
4 Check
Let's check with a calculator: term number 1000 is about 1, and the next is about −1.
Still jumping.
Right!
n = 1000: an ≈ 1
Let's look back.
First: the first terms.
Then: divide by the highest power.
Then: decide.
Last: check.
Answer
= diverges