Does Σ n^2/(n^3 + 7) converge

Step by step

The question

Question: decide if the series converges: the sum of n² over (n³ + 7) for n from 1 to infinity.

1 For big n

For big n, only the highest powers matter, top and bottom.

Cancel: about 1 over n.

2 The known series

We know the sum of 1/n: a p-series with p = 1.

p is 1, so it diverges.

3 Compare

The terms are not simply smaller or bigger here.

So use the limit comparison test: divide aₙ by bₙ and let n grow.

Dividing by 1/n is the same as times its flip.

Top and bottom now have the same highest power with the same number: the limit is 1.

The limit is a positive number.

Then the two series do the same thing: both converge, or both diverge.

4 Decide

So our series diverges.

Careful!

Smaller than a series that diverges tells us nothing.

We need bigger than a diverging one, or the limit test.

5 Check

Let's check with a calculator: 100, 1000 and 10000 terms add up to about 3.96, 6.26 and 8.56.

It keeps growing.

Right!

Let's look back.

First: for big n.

Then: the known series.

Then: compare.

Then: decide.

Last: check.