Step by step
The question
Question: find the sum 9 − 3 + 1 − 1/3 + and so on.
The dots at the end mean: it goes on forever.
1 Find the ratio
2 Look at the sums
1/3 is less than 1.
Each new term is smaller, and the sign flips.
So the sums jump up and down, but less each time: they settle down near one number.
Let's add the terms one at a time, and write each running sum: 9, 6, 7, and so on.
3 Use the sum rule
In the sum rule, r to the power n gets closer and closer to 0.
So it drops out, and the sum is the first term over 1 minus r.
Careful!
r is negative.
1 minus a minus is 1 plus: 4/3, not 2/3.
Put in a₁ = 9 and r = −1/3.
1 + 1/3 is 4/3.
So we divide by 4/3.
Dividing by 4/3 is times its flip, 3/4.
So the sum is 27/4.
4 Check
Let's check: we keep adding terms.
The running sums are 6.741, 6.753, 6.749.
They get closer and closer to 27/4.
It works!
Let's look back.
First: find the ratio.
Then: look at the sums.
Then: use the sum rule.
Last: check.