Does Σ 5/(n^3 + n) converge Step by Step

Step by step

The question

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

Σ_{n = 1}^∞5n³ + n =

1 For big n

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

a_n = 5n³ + n ≈ 5n³

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, 5, does not change that.

Σ_{n = 1}^∞1n³: p = 3 > 1, converges

3 Compare

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

n³ + n > n³

A bigger bottom makes a smaller fraction: every term is smaller than 5/n³.

a_n < 5n³

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

So ours add up to a number too.

smaller than a converging series: converges

4 Decide

So our series converges.

= converges

Careful!

Smaller than a series that diverges tells us nothing.

Compare with one that converges.

Careful: not a_n < 1n, so it diverges ✗

5 Check

Let's check with a calculator: 100, 1000 and 10000 terms add up to about 3.3591, 3.3593 and 3.3593.

It settles.

Right!

S₁₀₀ ≈ 3.3591, S₁₀₀₀ ≈ 3.3593, S₁₀₀₀₀ ≈ 3.3593

Let's look back.

First: for big n.

Then: the known series.

Then: compare.

Then: decide.

Last: check.

Answer