We have discussed whether a series will converge, when for each . But what about when we let be anything?

What happens to ? When we expand it out, we see that it looks like . But by comparison for each . And we *know* the series converges while diverges. So what happens to ?

**1. Proposition.** Let

(1)

be a given series where . Then the series converges if

- for each ;
- .

*Proof:* We have in the sequence of partial sums

(2)

so

(3)

But we also have

(4)

So

(5)

and

(6)

Conclusion: . ▮

**Example 1.** * Consider the series *

(7)

We see that for each , and

(8)

Thus the Alternating Series Test implies the alternating Harmonic series converges.

**Example 2.** * Consider the series *

(9)

We see

(10)

why? Well, multiply both sides by and we get

(11)

Cross multiplication gives us

(12)

Subtracting from both sides gives us

(13)

So our series satisfies the first condition for the alternating series test.

*
**
We also see that *

(14)

for “large ”. So we see

(15)

Thus our series satisfies the criteria for the alternating series test, which implies convergence.

**2. Definitions.** A series is **“Absolutely Convergent”** (or *“Converges Absolutely”*) if converges.

On the other hand, if is divergent, then we call the series **“Conditionally Convergent”** (or we say the series *“Converges Conditionally”*).

**Example 3.** * We see that is conditionally convergent, since diverges. *

**Example 4.** * The series is absolutely convergent since the integral test tells us converges. *

*Question:* let be a convergent series, and for each . Does the series converge?

Stop and think before continuing!

**3. Absolute Convergence Test.** If conveges, then converges.

*Proof:* We see first that

(16)

So what? Well, we see that

(17)

Since converges, we see converges too. But by the comparison test, we see

(18)

converges. So what? We haven’t proven converges, have we? Consider the following trick

(19)

Therefore the series converges. ▮

**Example 5.** * Consider the series *

(20)

What to do? We know

(21)

So what? We know converges by the integral test. Then

(22)

converges by comparison. By the absolute convergence test, we know our series in Equation (20) converges.

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## About Alex Nelson

I like math. I like programming. Most of all, I love puzzles.