Which is true about the series Σ from n = 1 to infinity of (n + 4)/n? Step by Step

Step by step

The question

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

Σ_{n = 1}^∞n + 4n

There are four choices.

First we solve it, the normal way.

Then we pick.

1 The term

A series adds up the terms aₙ for ever.

First look at one term, (n + 4) over n.

a_n = n + 4n

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 = 1 + 4n1

Every part with n at the bottom goes to 0.

as n → ∞ = 1 + 01

So the terms go to 1.

= 1

3 Decide

The terms do not go to 0.

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

diverges

Careful!

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

The sum grows by about 1 every time.

Careful: not the sum is 1 ✗

4 Check

Let's check with a calculator: the sums of the first 10, 100, 1000 terms are 21.72, 120.75, 1029.94.

They keep growing.

Right!

S₁₀ ≈ 21.72, S₁₀₀ ≈ 120.75, S₁₀₀₀ ≈ 1029.94

5 Pick the choice

Our answer is diverges.

That is choice D.

D) diverges

6 Why not the others

A is converges.

Wrong: the terms go to 1, not 0.

A) converges ✗ the terms go to 1, not 0

B is converges to 4.

Wrong: the sum keeps growing.

B) converges to 4 ✗ the sum keeps growing

C is converges to 1.

Wrong: the terms go to 1: the sum keeps growing.

That is the Careful mistake from before.

C) converges to 1 ✗ the terms go to 1: the sum keeps growing

Let's look back.

First: the term.

Then: the limit.

Then: decide.

Then: check.

Then: pick the choice.

Last: why not the others.

Answer