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Section 10.2
Sequences, Series, and Convergence:

Zooming in on the idea of a sequence of
values that approximate a value of a function

Terminology

A sequence is an ordered list of numbers. It is sometimes called an indexed list. Each number in the list is indexed by its ordinal position in the list. Another way to state this is that a sequence is a function from a subset of $\mathcal{N}$, the set of natural numbers $\{1, 2, 3, ...\}$, to the real numbers.

When the subset of $\mathcal{N}$ contains numbers from 1 to some definite number, the sequence is finite. When the subset is all of $\mathcal{N}$, the sequence is infinite.

The function f defined as $f(n)=2n-1, n \in \mathcal{N}$ produces the sequence $\left\langle f(1), f(2), f(3), \cdots \right\rangle=\left\langle 1, 3, 5, \cdots\right\rangle$.

The terms in a sequence need not follow a pattern. The list $\langle 2, 45, 30, 77, 37, 88, 100, 65\rangle$ in pointed brackets is a sequence simply because its terms are ordered. The list's 1st term is 2, its 2nd term is 45, its 3rd term is 30, and so on. The pointed brackets are used to indicate that the list is ordered. If you change term's order in a sequence, then you've made a different sequence.

The difference between a set of terms and a sequence of terms is more easily understood by a shopping example. If on a shopping trip you write a reminder to get sugar, flour, peanut butter, Pepsi, coffee, almond milk, and cereal, and you care only that you walk out of the store with these items, then you are thinking of a set of items. The order in which you put items in your basket doesn't matter to you.

However, if you write your reminder so that it reflects an efficient route among the store's isles, then changing the order changes the route you'll take within the store. Your reminder is a sequence. The order in which you write the items on it matters in terms of the route you take to get them.

Another convention is to represent a list with a general term and an index, such as $\langle a_i\rangle$. This notation hides the underlying function. Using function notation, we would write $\left\langle a(i)\right\rangle$. You will need to use function notation to define a sequence when graphing it in GC.

We often represent terms in a sequence using an index, such as $a_2$ to represent the 2nd term in the sequence $\langle a_i\rangle$. This means the same thing as saying that we are talking about $a(2)$, where $a$ is a function defined on a subset of $\mathcal{N}$. Using the example already given, we can say $\langle a_i\rangle=\langle a_1, a_2, a_3, a_4, a_5, a_6, a_7, a_8\rangle$, where $a_1=2$, $a_2=45$, $a_3=30$, and so on. We could just as well represent this sequence as $\langle a(i)\rangle=\langle a(1), a(2), a(3), a(4), a(5), a(6), a(7), a(8)\rangle$

The index of a term tells its place in the list. The index "5" in $a_5$ tells us that this term is the fifth in the list, which means that it is preceded in the list by four terms. The statement $a_5=37$ means that 37 is the 5th term in the sequence $\langle a_i\rangle$. You could also mix notations, writing $s_5=s(5)$.

An infinite sequence is one that has no last term. It is a function defined on all of $\mathcal{N}$. For example, the notation $$\left\langle\frac{1}{2^i}\right\rangle_{i=1}^\infty$$represents the infinite sequence $\left\langle\dfrac{1}{2^1},\dfrac{1}{2^2},\dfrac{1}{2^3},\, ...\right\rangle$. It is also a shorthand for $$f(i)=\frac{1}{2^i}, i \in \{1, 2, 3, \cdots\}.$$

Reflection 10.2.1. Represent the sequence $\left\langle b_i\right\rangle=\left\langle \dfrac{1}{2}, \dfrac{2}{3}, \dfrac {3}{4}, \cdots\right\rangle$ using function notation.

A series is the sum of the numbers in a sequence. The series from the sequence $\langle a_i\rangle=\langle a_1, a_2, a_3, a_4, a_5, a_6, a_7, a_8\rangle$ stated above is $$\begin{align}\sum\limits_{i=1}^8 a_i&=a_1+a_2+a_3+a_4+a_5+a_6+a_7+a_8\\[1ex]&=2+45+30+77+37+88+100+65\end{align}$$

Many textbooks reserve the word "series" for the sum of terms in an infinite sequence. We will use "series" more broadly, to cover both finite and infinite sequences. In our usage, an "infinite series" is simply a series created from an infinite sequence.

The ideas of sequence and series can be mixed.You can create a sequence of partial sums from a sequence. This is just a list of series that you create by listing the sums of successively longer subsequences.

A sequence of partial sums $\left\langle S_i\right\rangle$ is made from a sequence $\left\langle s_i \right\rangle$ by making $S_k=\sum\limits_{i=1}^{k} s_i$.

For example, the sequence $\left\langle s_i\right\rangle=\left\langle\dfrac{1}{2^1},\dfrac{1}{2^2},\, ...\right\rangle$ can be turned into a sequence of partial sums $\left\langle S_i\right\rangle$ where the $k^\mathrm{th}$ term in $\left\langle S_i\right\rangle$ is $\displaystyle{\sum\limits_{i=1}^{k}\frac{1}{2^i}}$:
$$\begin{align}\text{Given that }\left\langle s_i\right\rangle &=\left\langle\dfrac{1}{2^1},\dfrac{1}{2^2},\dfrac{1}{2^3},\dfrac{1}{2^4},\, ...\right\rangle\text{, then}\\[1ex]
\left\langle S_i\right\rangle &=\left\langle \sum\limits_{i=1}^{1}\frac{1}{2^i},\sum\limits_{i=1}^{2}\frac{1}{2^i},\sum\limits_{i=1}^{3}\frac{1}{2^i},\, ...\right\rangle \\[1ex]
&=\left\langle \left( \frac{1}{2}\right),\left(\frac{1}{2}+\frac{1}{4}\right),\left(1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}\right),\left(\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}\right),\left(\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\frac{1}{32}\right)\,...\right\rangle\\[1ex]
&=\left\langle \frac{1}{2},\frac{3}{2},\frac{7}{4},\frac{15}{8},\frac{31}{16},\frac{63}{32},\, ...\right\rangle\end{align}$$
Figure 10.2.1 gives a way to visualize the construction of the sequence of partial sums given above.

Figure 10.2.1. Constructing a sequence $\left\langle S_i\right\rangle$ of partial sums by adding the terms
in successively longer subsequences of the sequence $\left\langle s_i\right\rangle$.

Reflection 10.2.2. Consider the sequence of partial sums $\left\langle S_i\right\rangle=\left\langle \frac{1}{2},\frac{3}{2},\frac{7}{4},\frac{15}{8},\frac{31}{16},\frac{63}{32},\, ...\right\rangle$ from the example immediately above. Use GC to calculate $S_{72}$, the $72^{nd}$ partial sum in $\left\langle S_i\right\rangle$. Produce a statement in GC that will calculate $S_k$, the $k^{th}$ partial sum in $\left\langle S_i\right\rangle$.

Reflection 10.2.3. Rewrite Reflection 10.2.2 so that it says the same thing, but uses function notation to say it. Rewrite your GC file that you made for Reflection 10.2.2, but use function notation if you didn't use function notation originally.

An infinite sequence converges if all terms after some place in the sequence become essentially indistinguishable from each other.

 An infinite series converges if its sequence of partial sums converges.

Reflection 10.2.4. Restate the definition of convergence of an infinite sequence so that it says the same thing, but uses function notation to say it.

The phrase "become essentially indistinguishable from each other" is a matter of context and purpose. You might think at first that terms of a sequence are essentially indistinguishable if they agree to 15 decimal places. Later, you might decide that terms must agree to 22 decimal places before you consider them to be essentially indistinguishable from each other.

We must say that while our criterion for convergence, that terms in a sequence become "essentially indistinguishable from one another", worked well for us up to this moment, it was largely because we worked with integrals over finite intervals. We shall soon see that that we must think more carefully about what we mean that terms in a sequence become essentially indistinguishable from one another.

Exercise Set 10.2.1

(lots of practice with sequences and sums: expanding, representing general terms, evaluating in GC, graphing in GC.

Graphing sequences

At the end of Section 10.1, we created a table of values to inspect whether $A_\mathrm{quad}(1,3,\Delta x)$ converged as $\Delta x$ became smaller. Figure 10.2.1 does the same thing, but does it in a way that creates sequences of values. The first column, labeled n lists the numbers from 1 to 10, ini order. The second column, labeled $\Delta x=\frac{1}{n}$, lists values of $\Delta x$ that are created from values of n. The third column lists values of $A_\mathrm{quad}(1,3,\Delta x)$ that correspond to values of $\Delta x$.

We can consider each of these lists as a sequence.


Figure 10.2.2. Using a table to create the sequences
$$\begin{align}\left\langle n_i\right\rangle&=\left\langle i \right\rangle\\[1ex]
\left\langle \Delta x_i\right\rangle&=\left\langle\frac{1}{n_i}\right\rangle\text{, and}\\[1ex]
\left\langle A_i\right\rangle&=\left\langle A_\mathrm{quad}(1,3,\Delta x_i\right\rangle
\end{align}$$


Assuming that you have defined $A_\mathrm{quad}$ and its auxiliary functions, you can use GC to graph the values of $A_\mathrm{quad}(1,3,\Delta x_k)$ in relation to $x=3$ for $k=1$ to $k=100$ by typing the statement in Equation 10.2.1

which produces the graph in Figure 10.2.3.

Figure 10.2.3. GC's plot of points $(3, A_\mathrm{quad}(1,3,1/k))$ for $k\in\{1, 2, \cdots,100\}$.

Figure 10.2.3's plot of points associated with $(3, A_\mathrm{quad}(1,3,1/k))$ for $k\in\{1, 2, \cdots,100\}$ is not very useful for visual inspection. The plotted points overlap, and we cannot tell which point goes with which value of $k$. We can resolve that problem easily by plotting $A_\mathrm{quad}(1,3,1/k))$ against values of $k$, as in Figure 10.2.4.

Figure 10.2.4. Values of $(3, A_\mathrm{quad}(1,3,1/k))$ plotted agains values of k, $k\in\{1,2,\cdots,100\}$.

 Figure 10.2.4 suggests that the sequence $\left\langle A_k\right\rangle$=$\left\langle A_\mathrm{quad}(1,3,1/k)\right\rangle$ converges quickly so that all terms past $A_5$ appear essentially indistinguishable. However, this is merely a matter of scale.

The sequence in Figure 10.2.4, magnified vertically around the points $(k,A_\mathrm{quad}(1,3,1/k)$ for $k=45$ to $k=100$ (Figure 10.2.5), shows otherwise. At the level of vertical magnification in Figure 10.2.5, values of $A_\mathrm{quad}(1,3,1/k)$ are clearly distinguishable. We shall revisit this issue in the next section when we take a closer look at the idea of convergence.

Figure 10.2.5. A closer look at values of $A_\mathrm{quad}(1,3,1/k)$ in relation to values of $k\in \{45, 46, \cdots, 100\}$.

(Do the same with geometric sequence and harmonic sequence)


Convergence (revisited)

(Fragmentary thoughts)

(A sequence converges if given any level of vertical magnification, there is a place in the sequence where all terms past that one are essentially indistinguishable. State this using index notation and function notation. Use figures from class lectures that show a window height as epsilon.)

(Consider the function $r_f$ defined as $r_f(x)=1+1/3^x$, where $x$ varies through the real numbers greater than or equal to 1. Think of the sequence $f(n)=1+1/3^n$ as an approximate rate of change function with $\Delta x=1$. [Graph]. A series is like an accumulation function that has the sequence as its rate of change function that is constant over $\Delta x$-intervals of length 1. [Graph]. We can  compare the sum of the sequence to the integral.

If f is a decreasing function, f(x)>0, use the right method to compare the series with the integral of f. If the integral of f has a limit, the series has a limit.

Think of necessary conditions for convergence of a sequence's series. For sure, terms must become essentially equal to 0.