Step by step
The question
Question: find the limit of the sequence a n = (−1)ⁿn/(n + 1), as n grows.
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.
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.
As n grows, 1/n gets as small as we like: it goes to 0.
So does any number over a power of n.
3 The limit
Every part with n at the bottom goes to 0.
What is left: 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.
4 Decide
There is no single number at the end: the sequence diverges.
Careful!
Converges means the terms settle on one number.
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!
Let's look back.
First: the first terms.
Then: divide by the highest power.
Then: the limit.
Then: decide.
Last: check.