Step by step
The question
Question: decide if the series converges: the sum of 1/n⁴ for n from 1 to infinity.
1 The function
Make a function from the term: put x in place of n.
The series adds f(1), f(2), f(3) and so on.
For x from 1 on, it is positive, and it gets smaller: a bigger x makes a bigger bottom.
The integral test: the series and the area under f from 1 to infinity do the same thing.
Both are finite, or both are infinite.
2 The integral
Infinity is not a number: stop at t, then let t grow, like improper integrals.
The power rule: add 1 to the power, divide by the new power.
3 Let t go
Put in t, then take away the same with 1.
As t grows, this part goes to 0.
So the area is 1/3: a number.
4 Decide
The area is finite, so the series converges too.
This gives a rule for every power of n: the p-series rule.
Here p is 4, bigger than 1.
Careful!
The area tells us only that the series converges.
The sum itself is a different number.
5 Check
Let's check with a calculator: the first 1000 terms add up to about 1.082.
It stays below the first term plus the area, 1.333.
It settles.
Right!
Let's look back.
First: the function.
Then: the integral.
Then: let t go.
Then: decide.
Last: check.