Does Σ 2/(n^3 + 5) converge

Step by step

The question

Question: decide if the series converges: the sum of 2/(n³ + 5) for n from 1 to infinity.

1 For big n

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

2 The known series

We know the sum of 1/n to the 3: a p-series with p = 3.

p is bigger than 1, so it converges.

A number in front, 2, does not change that.

3 Compare

The bottom has extra plus terms: it is bigger than n³.

A bigger bottom makes a smaller fraction: every term is smaller than 2/n³.

The direct comparison test: our terms are smaller than terms that add up to a number.

So ours add up to a number too.

4 Decide

So our series converges.

Careful!

Smaller than a series that diverges tells us nothing.

Compare with one that converges.

5 Check

Let's check with a calculator: 100, 1000 and 10000 terms add up to about 0.6264, 0.6265 and 0.6265.

It settles.

Right!

Let's look back.

First: for big n.

Then: the known series.

Then: compare.

Then: decide.

Last: check.