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1d
comment Uniform convergence of series $\sum\limits_{n=2}^\infty\frac{\sin n x}{n\log n}$
@DavideGiraudo As a separate question, do you know any way to argue that $n a_n \rightarrow 0$ is also a necessary condition for uniform convergence?
1d
comment A Result About Sequences And Series
I would also like elaboration on this hint. I can't figure out how to apply it. Please explain more.
1d
comment $S(x)=\sum_{n=1}^{\infty}a_n \sin(nx) $, $a_n$ is monotonic decreasing $a_n\to 0$: Show uniformly converges within $[\epsilon, 2\pi - \epsilon]$
Interesting to see how Dirichlet did it. Another way: $\sin(kx)=Im(e^{ikx})$ so $\Sigma_{k=1}^N sin(kx)= Im \left(\sum_{k=1}^N e^{ikx} \right)$ which by Geometric Series formula is $Im ( \frac{1-e^{ix(N+1)}}{1-e^{ix}} )$ which, maximizing the top, is $\le \frac{2}{|1-e^{ix}|}$. Now $|1-e^{ix}|$ is continuous with zeros only at the endpoints of $[0, 2\pi]$. So on compact subsets, it attains a uniform minimum $>0$, which gives us a uniform finite bound $M_\epsilon$ on $[\epsilon, 2\pi-\epsilon]$.
2d
comment Showing that rationals have Lebesgue measure zero.
Using the countable additivity of Lebesgue measure doesn't seem in the spirit of the problem, or the example proofs given. The originally approach does in fact work, with slight modification. Notice we run into the problem that $\sum_{n=1}^\infty 2\epsilon =\infty$. We need a convergent sum. The fix clearly presents itself: simply take the intervals to be $(q_n-\epsilon_n, q_n+\epsilon_n)$ where $\epsilon_n=\epsilon\cdot 2^{-n}$. Then $\sum_{n=1}^\infty \epsilon 2^{-n}=4\epsilon$, which now goes to $0$ with $\epsilon$.
Jul
2
awarded  Curious
Jun
9
awarded  Popular Question
May
9
comment How does Hilbert's Nullstellensatz generalize the “fundamental theorem of algebra”?
It is usually the "weak Nullstellensatz" that is referred to as a multivariable analogue of FTA. For $k$ an algebraically closed field the "weak Nullstellensatz" can be stated as: "For any proper ideal $I$ in $k\left[X_1, X_2, \ldots, X_n\right]$, $V(I)$ is non-empty." Since ideals in $k[X]$ are principle, the $1D$ analogue contains precisely the same content as FTA. A proper ideal $I$ in $k[X]$ is given by a non-constant polynomial with zeros $V(I)$.
May
9
revised What is the completion of this space?
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May
9
comment Advice on taking a Course in Logic.
It might be worth replacing the word "logic" everywhere with "mathematical logic". As you can see from this comment section, there is a big difference. For example people who study plain "logic" are capable of saying things like "set theory really isn't logic".
May
9
revised What is the completion of this space?
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May
9
answered What is the completion of this space?
May
9
revised Prove that $\{n\}$ is a Cauchy sequence that doesn't converge.
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May
9
answered Prove that $\{n\}$ is a Cauchy sequence that doesn't converge.
May
9
comment Prove that $A\subseteq B\Longleftrightarrow A\cap B = A$
Unless stated otherwise, $\subseteq$ and $\subset$ mean the same thing. To avoid ambiguity, $\subsetneq$ is used for strict containment. If an author wishes to reserve $\subset$ for strict containment, it should be stated explicitly.
May
9
revised Proof strategy for $(\Leftarrow)$: If $g \circ f = id_A$, then $f$ onto $\Leftrightarrow$ $g$ 1-1. [Chartrand 3Ed P239 9.72]
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May
9
answered Proof strategy for $(\Leftarrow)$: If $g \circ f = id_A$, then $f$ onto $\Leftrightarrow$ $g$ 1-1. [Chartrand 3Ed P239 9.72]
May
9
answered Number of Binary Operations On a Set
May
9
comment The epsilon-delta definition of continuity
The reverse definition does not allow $\delta$ to depend upon $x$. The statement is that a single $\delta$ should work for all $x$. Any function with bounded range, for example, would fail the reverse condition.
May
9
answered Evaluating the reception of (epsilon, delta) definitions
May
9
revised How did Newton and Leibniz actually do calculus?
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