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

Step by step

The question

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

Σ_{n = 1}^∞1n² + n =

1 For big n

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

a_n = 1n² + n ≈ 1n²

2 The known series

We know the sum of 1/n to the 2: a p-series with p = 2.

p is bigger than 1, so it converges.

Σ_{n = 1}^∞1n²: p = 2 > 1, converges

3 Compare

The bottom has extra plus terms: it is bigger than n².

n² + n > n²

A bigger bottom makes a smaller fraction: every term is smaller than 1/n².

a_n < 1n²

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 0.9901, 0.999 and 0.9999.

It settles.

Right!

S₁₀₀ ≈ 0.9901, S₁₀₀₀ ≈ 0.999, S₁₀₀₀₀ ≈ 0.9999

Let's look back.

First: for big n.

Then: the known series.

Then: compare.

Then: decide.

Last: check.

Answer