Articles of reference request

On the possible values of $\sum\varepsilon_na_n$, where $\varepsilon_n=\pm1$ (i.e., changing signs of the original series)

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 […]

mathematical books

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 […]

Proof of the extreme value theorem without using subsequences

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 […]

Advice about taking mathematical analysis class

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 […]

Are hyperoperators primitive recursive?

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 […]

Understanding measures on the space of measures (via examples)

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 […]

Homology of the loop space

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 […]

Ramanujan's False Claims

“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 […]

Books with similar coverage to Linear Algebra Done Wrong

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 […]

How many digits of the googol-th prime can we calculate (or were calculated)?

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 […]