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

Step by step

The question

Question: decide if the series converges: the sum of (n² + 1)/(3n²) for n from 1 to infinity.

Σ_{n = 1}^∞n² + 13n² =

1 The term

A series adds up the terms aₙ for ever.

First look at one term, (n² + 1)/(3n²).

a_n = n² + 13n²

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 to the 2, the highest power at the bottom.

a_n = 1 + 1n²3

Every part with n at the bottom goes to 0.

as n → ∞ = 1 + 03

So the terms go to 1/3.

= 13

3 Decide

The terms do not go to 0.

Adding numbers close to 1/3 for ever never settles: the series diverges.

= diverges

Careful!

1/3 is where each term ends up, not the sum.

The sum grows by about 1/3 every time.

Careful: not the sum is 1 over 3 ✗

4 Check

Let's check with a calculator: the sums of the first 10, 100, 1000 terms are 3.85, 33.88, 333.88.

They keep growing.

Right!

S₁₀ ≈ 3.85, S₁₀₀ ≈ 33.88, S₁₀₀₀ ≈ 333.88

Let's look back.

First: the term.

Then: the limit.

Then: decide.

Last: check.

Answer