@user21820 Yeah, it made sense. Unfortunately, it seems that I didn't express the purpose of my question very well and lots of answerers misinterpreted it.
I was curious to know what the following limit is:
$$\lim_{x\downarrow-1}\sum_{n=1}^\infty p_nx^{n-1}=\lim_{x\downarrow-1}(2+3x+5x^2+7x^3+11x^4+\dots)$$
where $p_n$ is the $n$th prime.
I graphed the first 6 or so partial sums:
but they converge terribly slow. WolframAlpha doesn't seem to ha...
Previous question: Alternating prime series
I've been looking at the following function, defined on $\Re(s)>0$.
$$\mathcal P^\star(s)=\sum_{n=1}^\infty\frac{(-1)^{n+1}}{(p_n)^s}$$
where $p_n$ is the $n$th prime.
This function is clearly analytic on $\Re(s)>0$, and so I was curious enough to t...
Haha, yeah. Unfortunately I've lost my motivation to answer probability/combinatorics question (they get rather tedious to explain), and those made up a large chunk of my past answers.
Don't delete it. Here's how I think about it: the points you get for stupid answers make up for the points you don't get for really good but unnoticed answers to questions that are difficult for the layperson to understand.
This is highly non-trivial. Note that in the series definition of $\Phi_n(x)$, the summand is $\frac{1}{F_{k}F_{k+n}}$, not $\frac{1}{F_{ak}F_{ak+n}}$.
@Nilknarf Okay. Well my answer does address your intended question, if my claim (uniform computability of elementary integrals over elementary bounds) holds. Unfortunately, I do not know enough to verify that claim.
@Nilknarf Same for me. Normal days are good. I like predictable things. =)
The existing answers are about groups, but here is a more general perspective.
Suppose we have some kind of structure $S$ with a substructure $T$, meaning that $T$ inherits the operations and predicates from $S$ and is closed under those operations. We could ask the general question of whether...
And I have no idea why someone downvoted it.
It explains what I consider to be the underlying reason for why we need normality to form the quotient group.
@LeakyNun I'm sure it was clear to me when I wrote that, but it's not to me now. It's supposed to be because x~e since x∈T. I thought I had specified that T contains the identity, but I can't find it.
I used e∈T in the second part of that paragraph as well.
But I should edit to make that clear, since it has successfully confused you and me for a while.
@LeakyNun Thanks for helping; is the edited version better now? =)
In fact, the language underlying S,T does not even need to have the constant for the identity e, because we can prove uniqueness of identity in T, so T cannot have a different identity from that of S.
@LeakyNun In my experience of teaching, there is no difference between the two. Anyone who properly understands one properly understands the other too.
exists.elim: let $\varphi$ and $\psi$ be formulas with $n$ being the only free variable. Then, $[\exists n \varphi] \to [(\forall n \varphi) \to \psi] \to \psi$
If you're working in a Hilbert-style system then yes there can be other free variables, which would be bound by outer contexts in a Fitch-style system.
Nope. I've been around much longer than you. In the past his username was "thegreatduck" and many many things occurred, and the moderators had to step in, at one point even suspending him for misrepresentation of facts.
I was not even involved then. I was just a silent drama observer.
I was curious to know what the following limit is:
$$\lim_{x\downarrow-1}\sum_{n=1}^\infty p_nx^{n-1}=\lim_{x\downarrow-1}(2+3x+5x^2+7x^3+11x^4+\dots)$$
where $p_n$ is the $n$th prime.
I graphed the first 6 or so partial sums:
but they converge terribly slow. WolframAlpha doesn't seem to ha...
The existing answers are about groups, but here is a more general perspective.
Suppose we have some kind of structure $S$ with a substructure $T$, meaning that $T$ inherits the operations and predicates from $S$ and is closed under those operations. We could ask the general question of whether...
