murray
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 Oct 10 comment How does Dirichlet regularization of $1 + 2 + 3 + …$ work? That does not answer my question: what alteration does one make to the terms of the series, and what limit is then taken (and how) -- by the method of DIRICHLET REGULARIZATION -- so as to assign a value to the "sum" of the given divergent series? Oct 8 comment How does Dirichlet regularization of $1 + 2 + 3 + …$ work? But just how does one analytically continue $\sum_{n=1}^{\infty} a_n^{-s}$, that is, how does one obtain a formula that one can then evaluate at $s=-1$? Oct 8 comment How does Dirichlet regularization of $1 + 2 + 3 + …$ work? I'm trying to understand exactly what one does to the series $\sum_{k=1}^{\infty} k$ so as to produce a series having a parameter $s$ form which some kind of limit produces value $-1/12$. Oct 8 revised How does Dirichlet regularization of $1 + 2 + 3 + …$ work? Completely recast question Oct 8 comment How does Dirichlet regularization of $1 + 2 + 3 + …$ work? @Almentos: Which particular formula do you refer to that I should see agrees with $\sum_{k=1}^{\infty} k^{-s}$ when $\Re(s) > 1$? Oct 3 comment How does Dirichlet regularization of $1 + 2 + 3 + …$ work? OK, then I rephrase: just what series is the Dirichlet regularization of $\sum_{k=1}^{\infty} 1/k^s$ and why is the value of that regularization (not the original series itself, of course) equal to $-1/12$ when $s =-1$? Oct 3 asked How does Dirichlet regularization of $1 + 2 + 3 + …$ work? Aug 10 answered How can I prove the integral $\int_{1}^{x} \frac{1}{t} \, dt$ is $\ln x$ with this approach? Aug 10 comment Let $f(x)=(x-a)g(x)$. Calculate $f '(a)= g(a)$. Write the definition of the derivative $f'(a)$ of $f$ at $a$ in terms of a limit. And for that, first look at what the difference quotient $\frac{f(a+h) - f(a)}{h}$ is. Jul 19 comment Let $A,B$, and $C$ be sets. If $A\subseteq B$, $B\subseteq C$, and $C\subseteq A$, then $A=C$. I'm trying to hint that you may already know a little proposition about arbitrary sets $X$ and $Y$ that reduces the proof of what you want to just invoking that proposition. What propositions do you already know about the subset relation? Jul 19 comment Compute Surface Integral As @Ivo Terek indicates in his (accepted) answer, when you integrate a scalar field over a surface, you do not take its dot product with the normal—as indeed you note would make no sense. Jul 19 comment Tangent plane and tangent lines to curves through a point Can you offer of such an example where there are no such smooth curves?? Jul 18 comment Show the function is continuous in $\Bbb R^2$ To compute the partial derivatives at $(0,0)$, you'll need to go back to their definition in terms of limits—and this would seem to be no simpler than dealing showing $\lim_{(x,y) \to (0,0) f(x, y) = 1$. Jul 18 comment Compute Surface Integral If the question is to integrate $x^2 + y^2$ then you are not integrating a vector field but rather the scalar field $F(x, y, z) = x^2 + y^2$. Jul 18 comment Let $A,B$, and $C$ be sets. If $A\subseteq B$, $B\subseteq C$, and $C\subseteq A$, then $A=C$. Do you already have at your disposal the little result that if $X \subset Y$ and $Y \subset X$, then $X = Y$? Jul 18 comment If we have an embedding $f:X \rightarrow A$, where $A \subset Y$, do we have to show $f^{-1}$ is continuous? "Embedding" may be used in more restrictive ways in a particular context, e.g., an isometric embedding. But if all you have are topological spaces, or all you're considering is topology, then "embedding" is synonymous with "topological embedding". Jul 18 comment Does this intuition for “calculus-ish” continuity generalize to topological continuity? As to your students' question as to why the projections should be continuous: Is that not a natural generalization of the situation the projections from the cartesian plane onto each of the coordinate axes? Jul 18 comment Does this intuition for “calculus-ish” continuity generalize to topological continuity? The issue for a function $f: X \to Y$ between topological spaces is: what is the meaning of $\lim_{x \to c} f(x)$? One may use the notion of "net" to formulate the definition and then, with that definition, the condition you state is equivalent to continuity at $c$, provided you state it a bit more precisely: for every net $(x_i)_{ \in I}$ that converges to $c$ in $X$, the net $\bigl((f(x_i)\bigr)_{i \in I}$ converges to $f(c)$ in $Y$. Jul 12 revised Tangent plane and tangent lines to curves through a point deleted 1 character in body Jul 12 revised Tangent plane and tangent lines to curves through a point added 10 characters in body