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--- math104-s22:s:jiayinlin:start [2022/02/04 06:57]
107.77.211.225 [Jan 27th]
+++ math104-s22:s:jiayinlin:start [2026/02/21 14:41] (current)
@@ Line 39: / Line 39: @@
 (This one is better explained just by a graph)
-[[HW]] is quite easy this week though, but cantor diagonal is a really amazing technique.
+[[HW3]] is quite easy this week though, but cantor diagonal is a really amazing technique.
+====Feb 11th====
+Series is no good for me.
+I spent my whole life on the last question in rudin in this week [[HW4]].
+Here are my 5 questions:
+In our proof, divergence in quotient and root test both implies an almost-geometric sequence in the tail. I really dont understand how is it helpful in determining most of the non-obvious diverging series, since we usually dont need to spend much effort to show an increasing sequence diverge. :(
+In hw I see for a positive sequence, the fact that its partial sum is a monotone sequence can be helpful to see its convergence. Is there a lot of instances that we can try to "zip" a sequence into positive parts, and then try to bound the partial sum? (its really just an intuition that may not be useful at all)
+When we try to show some convergence and divergence of some series expressed as fractions (i.e. $\frac{n+1}{n^3+1}$) can we just compare the power terms and say it is convergent? (since 1/n^2 is) or this only works for sequence.
+When we want to analyze the set of subsequential limits, it there anything else than limsup and liminf that we can do...?
+if we can show that a sequence is bounded by the max of two sequences that has divergent series, can we say the series diverges? (something like comparison test)
+====Feb 25th====
+MT went smooth although I made stupid mistakes.
+Metric space is really hard, even at the beginning. Here is my [[HW 5]], I spent a lot of time on Ross 13.17.
+It is really smart that someone reminded me of taking a Q inside each interval, since they are disjoint they are all unique, and thus there is a bijection.
+====Mar 9th====
+I have done [[HW 6]] but the last question I do not get it.
+I see that cantor set can be a valid counter example, but I still dont understand how the given statement is wrong. It seems a really valid bijection from every open interval to its left (infimum).
+====Mar 10th====
+[[hw7]] was easy, but the first question I think there must be better ways for it. Mine was literally all over the place...
+The question in class in interesting that every path-connected set is a connected set. I did not have time to get to that one during discussion, but my idea was that if the set $A$ is not connected, then $A=B \cup C$, B and C are clopen, nonempty and disjoint. Then take an element $x\in B$ and $y\in C$, if $f: [0,1]\to A$ satisfies $f(0)=x$ f is continuous, then $f^{-1}(B)$ is clopen and contains 0. However, we know [0,1] is connected, so the only nonempty clopen subset is itself, so $f([0,1])=B$, which means $f(1)\not=y$.
+====Mar 18th====
+[[hw 8]] is easy for me, but I dont quite get the Weierstrass-M test since for me I feel it is a too strong requirement that sup of all $|f_n|$ is bounded by something that sum up to converge then the the convergence is too obvious. I am concerned about what happens when $\sum\limits_m^n sup(|f(x)|)$ is not cauchy (you are adding up values for different x in each term) but $sup(\sum\limits_m^n |f(x)|)$ converge to 0 for all x and the series is still uniform cauchy and how we identify this sort.
+====Apr 9th====
+I did horribly in MT2, so I have to study harder. [[HW9]] is here.
+I did the first question wrong, but I figured out that integrate my answer will work. This one was really cool.
+{{:math104-s22:s:jiayinlin:pasted:20220415-164451.png}}
+This was my original answer I thought I was constructing some that vanish at both points
+{{:math104-s22:s:jiayinlin:pasted:20220415-164526.png}}
+This works now properly
+====Apr 17th====
+I had a really hard time going through R-S integral. I missed a class and I cannot follow anymore... :(
+I have been frantically going through Ross 35 but I found there are so many material.
+I did [[hw10]] with no ease, and I hope they are correct.
+====Apr 25th====
+This [[hw11]] is quite easy for me though the last one I took some time to come up with, because I am too far from being familiar with R-S integral.

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