andy reagan only the interesting stuff

Tough Week Recap

Mom and Pops, sorry I didn't get a chance to call today, I was doing HW then eating then reading at Mass and then figured y'all would be sleeping at 9:30...I'll try to call tomorrow. I was thinking about calling all day, isn't it the thought that counts?

I found this picture of the SU game this past break on my phone, I really like it!

Figured I'd talk about this week, since I've survived in good health and high spirits.  I still do have a hard exam coming up Tuesday, but I think that I can do well if I study a lot.  Whoops, I just looked and the midterm isn't till a week after that, on the 15th...we don't have class Tuesday! So that means I don't have class till 5PM...I'm thinking time for a long ride finally.

Anyway, I got B on the number theory test, which I was happy with.  I messed up a little on one problem (there were 6 hard proofs) so I can still feel good about it, had I not made that mistake I would've had an A! Class avg was a C.  I'm standing at a B in the class.  In Advanced Calc, I got an 83 on the test which was actually a little better than I thought I had done immediately after taking it, and it being one of the hardest 3000-levels at the university (supposively), I'll take that one too.  I have a 92 HW average which is great, so I can possibly pull a low A in the class.  I finished and handed in my numerical analysis computing project, which was a whole bunch of writing MATLAB polynomial interpolation algorithms, and it wasn't too too hard but took me a lot of time.  On Tuesday, I left the house near 7AM and didn't return until 10:30PM.  In the middle of the day, I had to decide between going on a training ride, d2 (food) or going to the job fair with the resumes I had printed out.  I ended up going to d2, since I do need to eat food, and then taking the extra time to work hard on my presentation for research that I had at 4.  The presentation went really well, and I'm super anxious to hear about whether or not I get the job offer this summer.   Really hoping for it.

Unfortunately, on Monday our intramural basketball team lost in the playoffs, but we have one more game tomorrow at 7:30.  I'm pretty pumped to shoot some hoops after the game...I know can out free throw the VT team...42%...

Wednesday night I finished welding the trike back together, which was my release for the day.  Thursday I did as much C++ programming as I could, to free my mind from MATLAB and LaTeX before the exam at 5PM.  I think the exam went well, I'll find out soon enough.  Actually I just went and it was up....83.  Not too pleased with that.  But it's only 10% of the grade, and the 6 projects are worth 50%, and I knocked out the one due Friday on Monday and got an 104 on it.  I got  an 104 on the first one too...I'll be okay.

In the upper right, you can see where the frame is broken, and the piece of metal I'm holding is what I added to reinforce it after fixing the crack

After the exam on Thursday, Dave and I went camping up above Mtn Lake at the War Spur overlook.  We set out for Wind Rock, but the fire road was pretty washed out with some boulders that I didn't end up risking my oil pan on.  People thought we were crazy, since it was raining for the first time in a long time, and was pretty freaking cold.  But out there hiking, it wasn't too wet and I was down to a t-shirt, carrying in my frame backpack on the trail.

backpack loaded to go camping

Friday morning is when I finished putting the trike together before class.  I actually rode it to class which was fun, and towed some people around.  After more delicious d2 for lunch (I went a total of 4 times this week somehow), I used the trike to deliver water bottles to the team, to East Coasters, and pick up Christian's truing stand.  That night, I went to build a wheel....and totally failed. I had made the order online with a hub in it, but Dave wanted to use his own hub so we just deleted the one online, but we bought a 32 hole rim, 33 spokes...and his hub was 36 spoke! I didn't realize it until I had the stand all set up, nipples out and was putting the spokes into the rim.

Set up to build the wheel...fail

Friday night I went to a salsa dancing thing that CRU put on, with Jessey, and had a lot of fun.  I was pleasantly suprised that I knew a bunch of people there...Bryan and Katie Matthews and basically everyone I know from CRU. And Olivia from Tri team too! I was really needing to finish the beer in my keg, and Stephan came over so kindly to help me out, we were planning on going downtown to grill and sell hot dogs from the trike.  I thought that there were at most a couple of beers left, so I opened the whole keg up, and uh-oh...it was slightly full.  Well, not full, but close to a gallon or something! It had felt so light.  Had I known, I should have recruited more help!! We managed to finish it...ohh so good black IPA...but didn't make it downtown lol.

Saturday was a lot of fun, I'm probably repeating a bunch from the previous posts.  After going to college gameday in the AM and getting some HW done (gameday was super cool, I got four carol lee donuts for free for brkfst!), Jeff and I started a day of building, brewing, and kegging with a trip on the trike to Eats for brewing supplies.  Jeff isn't a small guy by any means, but we made it there! I'm sure Main St traffic wasn't as appreciative, but oh well for them.  We managed to keg my Raison D'etre for Tuesday's competition, brew the IPA (well...maybe an I*amber*A really, it's not pale), and build a bench onto the trike.  Before the game, I made three trips from the parking lot below Lane up to Cassell...and I think I shoulda picked a path that wasn't up the side of a hill! The first group of took up, of two people, bet me that I couldn't get all of us and the huge bike up the hill...well I showed them wrong! The lady was a cyclist, and she was thoroughly impressed, said I must've had "legs of steel" cuz we were cruisin up the hill blowin by traffic and everybody walking.  I then took one guy up, which was a slight relief to legs, and went down for a final trip.  I talked a group of three into taking a ride up, and as a challenge! They didn't think I could do it either.... I stood and pedaled for everything I was worth all the way up that hill, and got all four us and the bike to the top!!  I would've loved to have had a power meter on that ride, we were cookin up the hill...and I was pretty close to collapsing.

The game was amazing, check out the video in the previous post!  And then today, I slept in (a lot) and have finished all of my work for the week!  To do my math HW, I usually write it out then type it into an email since I don't have a LaTeX compiler on the computer.  Sometimes, I just straight do the HW on the computer, but that's hard to do.  Here's what it looks like:

\item 2.7.1: Proving the Alternating Series Test (Thm 2.7.7) amounts to showing that the sequence of partial sums
\begin{equation*}
s_n = a_1 - a_2 + a_3 - \ldots \pm a_n
\end{equation*}
converges.  (The opening example in Section 2.1 includes a typical illustration of ($s_n$).)  Different characterizations lead to different proofs, and I will prove this by showing that $(s_n)$ is a Cauchy sequence.
\underline{Proof:} Let $(a_n)$ be a strictly decreasing sequence with $(a_n) \rightarrow 0$.
Since $(a_n) \rightarrow 0, \forall \epsilon > 0$ being given$, \exists N \in \mathbb{R}$ s.t. $\forall n > N, |a_n - 0| = |a_n| < \epsilon$.
Note that since $(a_n)$ is decreasing, this implies for $m \geq n$, $|a_{m}| < \epsilon$.
I will now show that sequence of partial sums $(s_n)$ is Cauchy.
For the same $N$ above, and for all $n \geq m > N$,
\begin{equation*} \begin{align} | s_n - s_m | & = | a_1 - a_2 + a_3 - \ldots \pm a_m \pm a_{m + 1} \pm \ldots \pm a_n - (a_1 - a_2 + a_3 - \ldots \pm a_m)\\
& = | \pm a_{m + 1} \pm a_{m + 2} \pm \ldots \pm a_n |\\
& \text{Taking the abs. value of each term, which can only be greater,}\\
& \leq | \pm | a_{m + 1} | \pm | a_{m + 2} | \pm \ldots \pm | a_n | |\\
& \text{And since, $m, n > N$ then}\\
& < | \pm \epsilon \pm \epsilon \pm \ldots \pm \epsilon|\end{align} \end{equation*}
If $(n - m)$ is even, then since the terms are alternating sign, this equals 0, which is clearly less than any $\epsilon > 0$.  If the difference (number of terms) is odd, then it becomes $| \pm \epsilon | = \epsilon$ and so $| s_n - s_m | < \epsilon \forall \epsilon > 0$, as claimed.//
\item 2.7.2(a): Provide the details for the proof of the Comparison Test (Thm 2.7.4) using the Cauchy Criterion for series.
Let $\epsilon > 0$ be given, and then since the greater sequence of $b_k$'s converges, take that $N$ and look at the $(m + 1)$ through $n$ expansion of $a_n$'s and this is less than or equal to the expansion of $b_n$'s which is less than $\epsilon$.
\item 2.7.6: (a) Show that is $\sum x_n$ converges absolutely, and the sequence $(y_n)$ is bounded, then the sum $\sum x_n y_n$ converges.
\underline{Proof:} Let $\sum x_n$ converge absolutely, and the sequence $(y_n)$ be bounded.  Since  $\sum x_n$ converges absolutely, we know that given $\epsilon > 0, \exists N \in \mathbb{R}$ s.t. $\forall n > m \geq N$
\begin{equation*} |x_{m + 1}| + |x_{m + 2}| + \ldots + |x_n| < \frac{\epsilon}{M} \end{equation*}
where $M$ is the bound of $(y_n)$.  Then for all $n > m \geq N$,
\begin{equation*} \begin{align*} & |y_{m + 1}x_{m + 1} + y_{m + 2}x_{m + 2} + \ldots + y_{n}x_{n}|\\
& \text{By the Triangle Inequality,}\\
& \leq |y_{m + 1}x_{m + 1}| + |y_{m + 2}x_{m + 2}| + \ldots + |y_{n}x_{n}|\\
& \leq |M x_{m + 1}| + |M x_{m + 2}| + \ldots + |M x_{n}|\\
& \text{Since M is positive,}\\
& = M\cdot |x_{m + 1}| + M\cdot |x_{m + 2}| + \ldots + M\cdot |x_{n}|\\
& = M\cdot ( |x_{m + 1}| + |x_{m + 2}| + \ldots + |x_{n}| )\\
& < M\cdot (\frac{\epsilon}{M}) = \epsilon \end{align*} \end{equation*}
So $x_n y_n$ converges, as claimed.//\\
(b) Find a counterexample that demonstrates that part (a) does not always hold if the convergence of $\sum x_n$ is conditional.
Let $x_n = \dfrac{(-1)^n}{n}$ and $y_n = (-1)^n$.  We have seen that $x_n$ converges conditionally, and $x_ny_n$ which is the harmonic series, does not converge.
\item 2.7.9 (Ratio Test): Given a series $\sum^{\infy}_{n = 1} a_n$ with $a_n \ne 0$, the Ratio Test states that if $(a_n)$ satisifies
\begin{equation*} \text{lim } | \frac{a_{n + 1}}{a_n} | = r < 1 \end{equation*}
then the series converges absolutely.
(a) Let $r'$ satisfy $r < r' < 1$.  (Why must such an $r'$ exist?)  Explain why there exists an $N$ such that $n \geq N$ implies $|a_{n + 1}| \leq |a_n| r'$.
Such an $r'$ must exists by the axiom of completeness, and the open interval with an upper bound of 1.
(b) Why does $|a_N| \sum (r')^n$ necessarily converge?
(c) Now, show that $\sum |a_n|$ converges.
\item 3.2.4) Prove the converse of Thm 3.2.5 by showing that if $x$ = lim $a_n$ for some sequence $(a_n)$ contained in $A$ satisfying $a_n \ne a$, then $x$ is a limit point of $A$.
\underline{Proof:} Let $x$ = lim $a_n$, $a_n \ne x, (a_n) \subset A$.
I will show $\forall \epsilon > 0, V_{\epsilon} (x)$ intersects $A$ at some point not equal to $x$.
Let $\epsilon > 0$ be given, and by definition, $V_{\epsilon} (x) = \{ y \in \mathbb{R} : |y - x| < \epsilon \}$.
Then $\exists N \in \mathbb{R}$ s.t. $\forall n > N$, $|a_n - x| < \epsilon$.
By hypothesis, $a_n \in A$ and $a_n \ne x$.  Also, since $|a_n - x| < \epsilon \Rightarrow a_n \in V_{\epsilon} (x)$.
$a_n (\ne x) \in V_{\epsilon} (x) \intersect A$.  Therefore, $x$ is a limit point of $A$.
\item 3.2.7) Let $x \in O$, an open set.  If $(x_n)$ is a sequence converging to $x$, prove that all but a finite number of the terms of $(x_n)$ must be contained in $O$.
\underline{Proof:} Since $O$ is an open set, $\exists V_{\epsilon} (x) \subset O$.  Let $\epsilon > 0$ be given, such that $V_{\epsilon} (x) \subset O$.  Since $x_n \rightarrow x$, $\exists N \in \mathbb{R}$ st $\forall n \in \mathbb{N}, n > N \Rightarrow |x_n - x| < \epsilon \Rightarrow x_n \in V_{\epsilon} (x)$.
So for $n > N$, $x_n \subset V_{\epsilon} (x) \subset O$.  Therefore, for at most $n$ terms is $(x_n) \nsubset O$.
\item 3.3.4) Show that if $K$ is compact and $F$ is closed, then $K \intersect F$ is compact.
\underline{Proof:} I will show that $K \intersect F$ is closed and bounded, and then by the HB Thm $K \intersect F$ is compact.
By the HB Thm, $K$ is closed and bounded. Let $x$ be a limit point of $K \intersect F$.  Then $x$ is also a limit point of $K$ and $F$.  Since $K$ is closed, $x \in K$.  Since $F$ is closed, $x \in F$.  Therefore, $x \in K \intersect F$.  So $K \intersect F$ is closed.
Since $K$ is bounded, let $M > 0$ be the bound of $K$.  Therefore, $a \in K \Rightarrow |a| \leq M \forall a \in M$.
$a \in K \intersect F \Rightarrow a \in K \Rightarrow |a| < M.$
Therefore, $K \intersect F$ is bounded.  As claimed.//
\item 2.7.1: Proving the Alternating Series Test (Thm 2.7.7) amounts to showing that the sequence of partial sums\begin{equation*}s_n = a_1 - a_2 + a_3 - \ldots \pm a_n\end{equation*}converges.  (The opening example in Section 2.1 includes a typical illustration of ($s_n$).)  Different characterizations lead to different proofs, and I will prove this by showing that $(s_n)$ is a Cauchy sequence.\underline{Proof:} Let $(a_n)$ be a strictly decreasing sequence with $(a_n) \rightarrow 0$.Since $(a_n) \rightarrow 0, \forall \epsilon > 0$ being given$, \exists N \in \mathbb{R}$ s.t. $\forall n > N, |a_n - 0| = |a_n| < \epsilon$.Note that since $(a_n)$ is decreasing, this implies for $m \geq n$, $|a_{m}| < \epsilon$.I will now show that sequence of partial sums $(s_n)$ is Cauchy.For the same $N$ above, and for all $n \geq m > N$,\begin{equation*} \begin{align} | s_n - s_m | & = | a_1 - a_2 + a_3 - \ldots \pm a_m \pm a_{m + 1} \pm \ldots \pm a_n - (a_1 - a_2 + a_3 - \ldots \pm a_m)\\& = | \pm a_{m + 1} \pm a_{m + 2} \pm \ldots \pm a_n |\\& \text{Taking the abs. value of each term, which can only be greater,}\\& \leq | \pm | a_{m + 1} | \pm | a_{m + 2} | \pm \ldots \pm | a_n | |\\& \text{And since, $m, n > N$ then}\\& < | \pm \epsilon \pm \epsilon \pm \ldots \pm \epsilon|\end{align} \end{equation*}If $(n - m)$ is even, then since the terms are alternating sign, this equals 0, which is clearly less than any $\epsilon > 0$.  If the difference (number of terms) is odd, then it becomes $| \pm \epsilon | = \epsilon$ and so $| s_n - s_m | < \epsilon \forall \epsilon > 0$, as claimed.//\item 2.7.2(a): Provide the details for the proof of the Comparison Test (Thm 2.7.4) using the Cauchy Criterion for series.Let $\epsilon > 0$ be given, and then since the greater sequence of $b_k$s converges, take that $N$ and look at the $(m + 1)$ through $n$ expansion of $a_n$s and this is less than or equal to the expansion of $b_n$s which is less than $\epsilon$.\item 2.7.6: (a) Show that is $\sum x_n$ converges absolutely, and the sequence $(y_n)$ is bounded, then the sum $\sum x_n y_n$ converges.\underline{Proof:} Let $\sum x_n$ converge absolutely, and the sequence $(y_n)$ be bounded.  Since  $\sum x_n$ converges absolutely, we know that given $\epsilon > 0, \exists N \in \mathbb{R}$ s.t. $\forall n > m \geq N$\begin{equation*} |x_{m + 1}| + |x_{m + 2}| + \ldots + |x_n| < \frac{\epsilon}{M} \end{equation*}where $M$ is the bound of $(y_n)$.  Then for all $n > m \geq N$,\begin{equation*} \begin{align*} & |y_{m + 1}x_{m + 1} + y_{m + 2}x_{m + 2} + \ldots + y_{n}x_{n}|\\& \text{By the Triangle Inequality,}\\& \leq |y_{m + 1}x_{m + 1}| + |y_{m + 2}x_{m + 2}| + \ldots + |y_{n}x_{n}|\\& \leq |M x_{m + 1}| + |M x_{m + 2}| + \ldots + |M x_{n}|\\& \text{Since M is positive,}\\& = M\cdot |x_{m + 1}| + M\cdot |x_{m + 2}| + \ldots + M\cdot |x_{n}|\\& = M\cdot ( |x_{m + 1}| + |x_{m + 2}| + \ldots + |x_{n}| )\\& < M\cdot (\frac{\epsilon}{M}) = \epsilon \end{align*} \end{equation*}So $x_n y_n$ converges, as claimed.//\\(b) Find a counterexample that demonstrates that part (a) does not always hold if the convergence of $\sum x_n$ is conditional.Let $x_n = \dfrac{(-1)^n}{n}$ and $y_n = (-1)^n$.  We have seen that $x_n$ converges conditionally, and $x_ny_n$ which is the harmonic series, does not converge.
\item 2.7.9 (Ratio Test): Given a series $\sum^{\infy}_{n = 1} a_n$ with $a_n \ne 0$, the Ratio Test states that if $(a_n)$ satisifies\begin{equation*} \text{lim } | \frac{a_{n + 1}}{a_n} | = r < 1 \end{equation*}then the series converges absolutely.(a) Let $r'$ satisfy $r < r' < 1$.  (Why must such an $r'$ exist?)  Explain why there exists an $N$ such that $n \geq N$ implies $|a_{n + 1}| \leq |a_n| r'$.Such an $r'$ must exists by the axiom of completeness, and the open interval with an upper bound of 1.(b) Why does $|a_N| \sum (r')^n$ necessarily converge?(c) Now, show that $\sum |a_n|$ converges.
\item 3.2.4) Prove the converse of Thm 3.2.5 by showing that if $x$ = lim $a_n$ for some sequence $(a_n)$ contained in $A$ satisfying $a_n \ne a$, then $x$ is a limit point of $A$.
\underline{Proof:} Let $x$ = lim $a_n$, $a_n \ne x, (a_n) \subset A$.I will show $\forall \epsilon > 0, V_{\epsilon} (x)$ intersects $A$ at some point not equal to $x$.Let $\epsilon > 0$ be given, and by definition, $V_{\epsilon} (x) = \{ y \in \mathbb{R} : |y - x| < \epsilon \}$.Then $\exists N \in \mathbb{R}$ s.t. $\forall n > N$, $|a_n - x| < \epsilon$.By hypothesis, $a_n \in A$ and $a_n \ne x$.  Also, since $|a_n - x| < \epsilon \Rightarrow a_n \in V_{\epsilon} (x)$.
$a_n (\ne x) \in V_{\epsilon} (x) \intersect A$.  Therefore, $x$ is a limit point of $A$.
\item 3.2.7) Let $x \in O$, an open set.  If $(x_n)$ is a sequence converging to $x$, prove that all but a finite number of the terms of $(x_n)$ must be contained in $O$.
\underline{Proof:} Since $O$ is an open set, $\exists V_{\epsilon} (x) \subset O$.  Let $\epsilon > 0$ be given, such that $V_{\epsilon} (x) \subset O$.  Since $x_n \rightarrow x$, $\exists N \in \mathbb{R}$ st $\forall n \in \mathbb{N}, n > N \Rightarrow |x_n - x| < \epsilon \Rightarrow x_n \in V_{\epsilon} (x)$.So for $n > N$, $x_n \subset V_{\epsilon} (x) \subset O$.  Therefore, for at most $n$ terms is $(x_n) \nsubset O$.\item 3.3.4) Show that if $K$ is compact and $F$ is closed, then $K \intersect F$ is compact.\underline{Proof:} I will show that $K \intersect F$ is closed and bounded, and then by the HB Thm $K \intersect F$ is compact.By the HB Thm, $K$ is closed and bounded. Let $x$ be a limit point of $K \intersect F$.  Then $x$ is also a limit point of $K$ and $F$.  Since $K$ is closed, $x \in K$.  Since $F$ is closed, $x \in F$.  Therefore, $x \in K \intersect F$.  So $K \intersect F$ is closed.Since $K$ is bounded, let $M > 0$ be the bound of $K$.  Therefore, $a \in K \Rightarrow |a| \leq M \forall a \in M$.$a \in K \intersect F \Rightarrow a \in K \Rightarrow |a| < M.$Therefore, $K \intersect F$ is bounded.  As claimed.//

I'll post up what that actually comes on looking like, it looks great.  It's actually the typesetting program that mathematicians use to write papers to submit them to journals, and probably what my textbook is written in.  Useful to know I suppose.  Here is what it comes out to.

That's all for now. This week: not as crazy, going to decide about spring break plans (possibly traveling home, we shall see).

Ciao!

PS: I'm going on a bike ride tomorrow at 3PM, check out the weather for then:

PPS: Bonus Pictures of the Backstreet's Pizza Challenge and others from my phone

the contest rules. it's never been done

the PIZZA. Jan and I had the closer one. it was thiiiick

We did work...but it's impossible. I informally resigned from competitive eating mid-competition

I did make Dean's List last semester, yeah!

And earlier this semester, I put cyclocross tires on the fixed gear, it's a beast commuting now

HOKIES

We beat Duke last night, and it was awesome.  We didn't even play that well, but played harder than we have this whole year.  I was right in the middle of this!!

http://www.youtube.com/watch?v=0TrlMYERvyM&feature=player_embedded

After getting out that giant sweaty mess, which was so crazy, then I got the trike and took Jeff, Rudy and Crowley on a parade downtown screaming and shouting with the crowds. Sooooo fun.  All of downtown was a pretty huge party.

I tied up the trike and went into the Cellar, the only place without a 50 person line, and Hayden and a bunch of Hillcrest people were there which was cool.  I was super thirsty from the rediculous workout I was getting so I went for a Schlitz as my celebration beer haha, $1.   It was a crazy night, watching the highlights shouting "lets go!!" and the rest of the Rivermill chanting back "HOKIESSS!!"