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\centerline{Mathematics 242 -- Principles of Analysis}
\centerline{Discussion 7 -- Infinite Series}
\centerline{November 20 and 27, 2000}
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\noindent
{\it Background}
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The last unit of this course will be devoted to the following
general question:  How can we tell whether an infinite sum of real numbers
$$\sum_{k=1}^\infty a_k\leqno(1)$$
``adds up'' or {\it converges} to a well-defined finite result. 
A sum with infinitely many terms is called an {\it infinite series}.
The numbers $a_k$ are called the {\it terms} of the series.

We will 
approach this question by the following method.  The thing that makes sense
immediately from the algebraic properties of the real number system
is to add a finite collection of numbers.  So we can stop after any finite
number $N$ of terms in the infinite sum above and consider the ``partial sum'':
$$s_N = \sum_{k=1}^N a_k = a_1 + a_2 + \cdots + a_N$$
In this way, we get a sequence of partial sums, whose behavior should
tell us whether the infinite sum (1) converges or not. 
Of course it is possible for an infinite sum {\it not} to add up 
to a finite result.  For example, if $a_k = 1$ all $k$, then
$S_N = {N(N+1)\over 2}$ which goes to $+\infty$ as $N\to\infty$.
We say that the series $\sum_{k=1}^\infty 1$ {\it diverges}.
On the other hand,
if $\lim_{N\to\infty} s_N$ exists and equals $S$, then we will say that
the series (1) converges to (intuitively, ``adds up to'') $S$.

For today, we will concentrate on the case where all $a_k \ge 0$.
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\noindent
{\it Discussion Questions}
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\item{A)} Assuming all $a_k \ge 0$, what can we say about the
sequence of partial sums $s_N$?  Under what conditions would
that sequence be ``guaranteed'' to converge to some finite
real value?  
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\item{B)}  In problem 15 from Section 1.4 you proved that 
$\sum_{k=1}^N {1\over k!} \le 2$ for all $N$.  What does this
say about the infinite series
$\sum_{k=1}^\infty {1\over k!}$.  Experiment with the numerical 
values for partial sums up to $N = 15$ and see if your 
statement seems reasonable.
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\item{C)}  In problem 11 from Section 1.4, you proved
that if $r$ is any real number, then the finite {\it geometric
series with ratio $r$} adds up to the following:
$$\sum_{k=0}^N r^k = {1-r^{N+1}\over 1-r}.$$
Suppose $r > 0$.
Thinking of the left hand side here as the partial sum for 
the infinite geometric series $\sum_{k=0}^\infty r^k$, when
will it be true that this series converges?
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\item{D)}  Consider the series $\sum_{k=1}^\infty {1\over k^2+k}$,
where $a_k = {1\over k^2+k}$ for all $k \ge 1$.
All partial sums of this series are rational numbers.  Compute the
partial sums $s_1,\ldots,s_8$ and look for a pattern.  Prove your
guess, and use that to determine an answer to our general question
in this case.  If $\sum_{n=1}^\infty {1\over k^2+k}$ does
add up to a finite result, what is the sum?
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\item{E)}
On the other hand, consider the series $\sum_{k=1}^\infty {1 \over k}$.
Consider the partial sums $s_2,s_4,s_8,s_{16}$ and so forth--the
partial sums where $N = 2^\ell$ is a power of $2$.  What can you say about
$s_{2^{\ell+1}} - s_{2^\ell}$?  Determine an answer to our general question
in this case.  If $\sum_{n=1}^\infty {1\over k}$ does
add up to a finite result, what is the sum?
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\item{F)}  Suppose you know that series $\sum_{k=1}^\infty a_k$ 
(all $a_k \ge 0$)
{\it converges\/}, and that the terms in a second
series $\sum_{k=1}^\infty b_k$ satisfy $b_k \le a_k$.
How are the $N$th partial sums for the $a$-series and  
the $b$-series related?  Must the $b$-series also converge? 
Prove your assertion.
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\item{G)}  Suppose you know that the $\sum_{k=1}^\infty a_k$ 
(all $a_k \ge 0$)
{\it diverges\/}, and that the terms in a second
series $\sum_{k=1}^\infty b_k$ satisfy $b_k \ge a_k$.
How are the $N$th partial sums for the $a$-series and  
the $b$-series related?  Must the $b$-series also diverge? 
Prove your assertion.
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\item{H)}  Suppose you know that the $\sum_{k=1}^\infty a_k$ 
(all $a_k \ge 0$)
{\it diverges\/}, and that the terms in a second
series $\sum_{k=1}^\infty b_k$ satisfy $b_k \le a_k$.  Can
you say anything in this case?
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\noindent
{\it Assignment}
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Group write-ups due in class Wednesday, November 29.
\bye

