I have used the following result in an answer on this site. Suppose that $a_n>0$ are positive real numbers such that $\sum\limits_{n=1}^\infty a_n=+\infty$ and $\lim\limits_{n\to\infty} a_n=0$. Ten for any choice of $A,B\in\mathbb R\cup\{\pm\infty\}$ such that $A\le B$ there exists a sequence $\varepsilon_n$ such that each $\varepsilon_n\in\{\pm1\}$, $$\liminf\limits_{n\to\infty} \sum_{k=1}^n \varepsilon_k a_k=A \qquad\text{ and }\qquad \limsup\limits_{n\to\infty} \sum_{k=1}^n […]

I am a Japanese so it is difficult for me to read English and I may make some grammatical mistakes. I have little experience with mathematics but plan to self-study mathematics by reading mathematics books in English. I would like to know what the best text is in each of the following subjects, set theory […]

I am preparing a lecture on the Weierstrass theorem (probably best known as the Extreme Value Theorem in english-speaking countries), and I would propose a proof that does not use the extraction of converging subsequences. I did not explain subsequences in my calculus course, and I must choose between skipping the proof of the theorem […]

I appologize if this isn’t the place to ask, if it’s not could you let me know and I will take it to meta? Anyway, so I am planning on taking a mathematical analysis course next spring, and I’m really excited about it because it seems so interesting and fun. However, I know this will […]

I apologize if this question is too basic. I have read that the Ackerman function is the first example of a computable but NOT primitive recursive function. Hyperoperators seem to be closely related to these functions, but I am not sure if they still keep the property of being NOT primitive recursive. My intuition is […]

Let $X$ be a Polish space. If it allows for an interesting answer, you may assume $X$ is compact or even $X=[0,1]$. The space $\mathcal{P}(X)$ of Borel probability measures on $X$ is also Polish (via the Prokhorov metric). Measures on $\mathcal{P}(X)$ (i.e. elements of $\mathcal{P}(\mathcal{P}(X))$) arise, for example in the ergodic decomposition. I’m looking to […]

Let $X$ be a nice space (manifold, CW-complex, what you prefer). I was wondering if there is a computable relation between the homology of $\Omega X$, the loop space of $X$, and the homology of $X$. I know that, almost by definition, the homotopy groups are the same (but shifted a dimension). Because the relation […]

“During his short life, Ramanujan independently compiled nearly 3900 results (mostly identities and equations). Nearly all his claims have now been proven correct, although a small number of these results were actually false and some were already known.” (Source) What are some of Ramanujan’s false claims? Note: A long time ago, I remember reading a […]

Axler’s book is great, but for my immediate purposes, it isn’t suitable. I’ve been looking at the Table of Contents of Linear Algebra Done Wrong by Treil starting at p. 5 of this document but there’s only one disadvantage to it: I hate learning from reading .pdfs online, and I’d rather not print the entire […]

Here, a lower and upper bound for the $n$-th prime are given. Applying the given bounds $$n(\ln(n\cdot\ln(n))-1)<p_n<n\cdot\ln(n\cdot\ln(n))$$ and the approximation $$p_n\approx n(\ln(n\cdot\ln(n))-1)+\frac{n(\ln(\ln(n))-2)}{\ln(n)}$$ we get that $p_{10^{100}}$ is somewhere between $2.346977\cdot 10^{102}$ and $2.35698\cdot 10^{102}$ and approximately $2.3471\cdot 10^{102}$ , so it has $103$ digits. How many digits can we determine of the googol-th prime with […]

Intereting Posts

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