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

Step by step

The question

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

Σ_{n = 1}^∞12ⁿ + 1 =

1 For big n

For big n, 2 to the n is huge: the + 1 hardly matters.

So the terms are about 1/2ⁿ.

a_n = 12ⁿ + 1 ≈ 12ⁿ

2 The known series

The sum of 1/2ⁿ is a geometric series: each term is 1/2 of the one before.

r is less than 1, so it converges.

Σ_{n = 1}^∞12ⁿ: r = 12 < 1, converges

3 Compare

The bottom of aₙ is bigger: it has 1 more.

2ⁿ + 1 > 2ⁿ

A bigger bottom makes a smaller fraction.

So every term is smaller than 1/2ⁿ.

a_n < 12ⁿ

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 0.7645, 0.7645 and 0.7645.

It settles.

Right!

S₁₀₀ ≈ 0.7645, S₁₀₀₀ ≈ 0.7645, S₁₀₀₀₀ ≈ 0.7645

Let's look back.

First: for big n.

Then: the known series.

Then: compare.

Then: decide.

Last: check.

Answer