## Series Convergence: What Does It Mean?

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 $\sum 2^{-n}$. We look at the partial sums, so, $S_{1}=1/2$, $S_{2} = 3/4$, $S_{3}=7/8$, $S_{4}=15/16$, and in general $S_{n}= (2^{n}-1)/2^{n} = 1-2^{-n}$.

We see we have a sequence $\{S_{n}\}$ and the sequence has its limit be
(1)$\displaystyle \lim_{n\to\infty}S_{n} = \lim_{n\to\infty} 1 - \frac{1}{2^{n}} = 1-\lim_{n\to\infty}2^{-n}= 1-0=1$.
We then define the series to converge to
(2)$\displaystyle \sum^{\infty}_{n=1}2^{-n}=1$.
This is our convention and definition, a series converges if and only if its sequence of partial sums converge.

3. Definition. Let $\displaystyle\sum^{\infty}_{n=1}a_{n}$ be given. Let $S_{n}=\sum^{n}_{k=1}a_{k}$ be the sequence of partial sums.

If $\displaystyle \lim_{n\to\infty}S_{n}=L$, we define $\displaystyle\sum^{\infty}_{n=1}a_{n}=L$ and say the series “Converges”.

If the sequence $\{S_{n}\}$ diverges, then the series $\sum a_{n}$ “Diverges”.

Example 1. Consider the series $\displaystyle\sum^{\infty}_{n=1}\cos(n\pi/2) = 0-1+0+1+\dots$. The sequence of partial sums does not “settle”: it diverges. So the series diverges.

Example 2. Consider the series
(3)$\displaystyle \sum^{\infty}_{n=1}\frac{6}{(2n+1)(2n-1)}=\sum^{\infty}_{n=1}\frac{6}{4n^{2}-1}=\sum^{\infty}_{n=1}\frac{3}{2n-1}-\frac{3}{2n+1}$.
We see that the partial sum is $S_{n} = 3 - 3/(2n+1)$. So
(4)$\displaystyle \lim_{n\to\infty}3-\frac{3}{2n+1}=3$.
Thus the series in Equation (3) converges to 3.

Theorem 1. If $\displaystyle\sum^{\infty}_{n=1}a_{n}$ converges, then $\displaystyle\lim_{n\to\infty}a_{n}=0$.

Proof. Let $\sum^{\infty}_{n=1}a_{n}=L$. Then $\lim S_{n}=L$ and $S_{n-1}=L$. We see $S_{n}-S_{n-1}=a_{n}$, so
$\displaystyle \lim_{n\to\infty}a_{n}=\lim_{n\to\infty}S_{n}-S_{n-1}=L-L=0$.
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 $\lim a_{n}\not=0$, then $\sum a_{n}$ diverges.”

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