Does Σ (4/3)^n converge

Step by step

The question

Question: decide if the series converges: the sum of (4/3)ⁿ for n from 1 to infinity.

1 The term

A series adds up the terms aₙ for ever.

First look at one term, (4/3)ⁿ.

The divergence test: if the terms do not go to 0, adding them for ever never settles.

The series diverges.

2 The limit

A number bigger than 1, multiplied by itself again and again, grows without end.

4/3 is bigger than 1: the terms grow without end.

3 Decide

The terms do not go to 0.

Adding numbers that grow for ever never settles: the series diverges.

Careful!

The limit of the terms tells us only about the terms, not about the sum.

4 Check

Let's check with a calculator: the sums of the first 5, 10, 20 terms are 12.86, 67.03, 1257.35.

They keep growing.

Right!

Let's look back.

First: the term.

Then: the limit.

Then: decide.

Last: check.