**1.** So we discussed what it means for a sequence to converge, and we are really interested in infinite series.

It makes sense to first ask “Does this series equal a finite number?” BEFORE we ask “What number is this series equal to?” So we construct a test for a series to see if it converges to a finite number.

**2.** Consider the series . We look at the partial sums, so, , , , , and in general .

We see we have a sequence and the sequence has its limit be

(1).

We then *define* the series to converge to

(2).

This is our convention and definition, a series converges if and only if its sequence of partial sums converge.

**3. Definition.** Let be given. Let be the sequence of partial sums.

If , we define and say the series **“Converges”**.

If the sequence diverges, then the series **“Diverges”**.

**Example 1.** Consider the series . The sequence of partial sums does not “settle”: it diverges. So the series diverges.

**Example 2.** Consider the series

(3).

We see that the partial sum is . So

(4).

Thus the series in Equation (3) converges to 3.

**Theorem 1.** If converges, then .

*Proof.* Let . Then and . We see , so

.

That concludes the proof. ▮

**4. Aside on Logic.** A proposition of the form “If *A*, then *B*” can give us two different propositions.

(1) The Contrapositive “If not *B*, then not *A*.” It is logically equivalent to “If *A*, then *B*.”

(2) The Converse “If *B*, then *A*.” This is not logically equivalent to “If *A*, then *B*.”

The contrapositive of Theorem 1 is “If , then diverges.”

Whenever you are handed a proposition, you should check its converse to see if its true or not.

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## About Alex Nelson

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

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