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Jan
25
comment How does Dirichlet regularization of $1 + 2 + 3 + …$ work?
Thank you, I'd like to see that explanation. My email address appears on: math.umass.edu/directory/emeritus-faculty/murray-eisenberg
Jan
21
comment How does Dirichlet regularization of $1 + 2 + 3 + …$ work?
What's the relevance of the 2nd formula that begins with the double sum $\sum{n=1}^{d\,p} f(n)\,\sum_{k=1}^{d-1}(e^{2i\pi k/d})^n$? And what is the purpose of that double sum — is taking a limit of that the very definition of Dirichlet regularization of the series $\sum_{n=1}^{\infty} f(n)$? Please help me by being precise and even prolix in your answer!
Jan
19
comment How does Dirichlet regularization of $1 + 2 + 3 + …$ work?
How precisely do you get the value of $\sum_{n=1}^{\infty} n\,c^n$ for values of $c$ that are around 1?
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?