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
We then define the series to converge to
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.
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.