Does Σ (2n - 1)/(3n + 2) converge Step by Step

Step by step

The question

Question: decide if the series converges: the sum of (2n − 1)/(3n + 2) for n from 1 to infinity.

Σ_{n = 1}^∞2n − 13n + 2 =

1 The term

A series adds up the terms aₙ for ever.

First look at one term, (2n − 1)/(3n + 2).

a_n = 2n − 13n + 2

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

The series diverges.

divergence test:

if a_n does not → 0,

the series diverges

2 The limit

Divide top and bottom by n, the highest power at the bottom.

a_n = 2 − 1n3 + 2n

Every part with n at the bottom goes to 0.

as n → ∞ = 2 − 03 + 0

So the terms go to 2/3.

= 23

3 Decide

The terms do not go to 0.

Adding numbers close to 2/3 for ever never settles: the series diverges.

= diverges

Careful!

2/3 is where each term ends up, not the sum.

The sum grows by about 2/3 every time.

Careful: not the sum is 2 over 3 ✗

4 Check

Let's check with a calculator: the sums of the first 10, 100, 1000 terms are 4.93, 63.22, 661.43.

They keep growing.

Right!

S₁₀ ≈ 4.93, S₁₀₀ ≈ 63.22, S₁₀₀₀ ≈ 661.43

Let's look back.

First: the term.

Then: the limit.

Then: decide.

Last: check.

Answer