Articles of analysis

Does the series $\sum_{n = 1}^{\infty}\left(2^{1/n} – 1\right)\,$ converge?

I’m trying to determine if the following sum converges or diverges (this is question 38 in section 11.7 of Stewart’s Early Transcendentals): $$\sum_{n = 1}^{\infty}(2^{1/n} – 1)$$ I’ve considered all of the techniques I know (integration test, ratio test, etc.), but I upon inspection, none of them will solve this problem. Any hints or help […]

definition of “weak convergence in $L^1$”

I have encountered two definitions of weak convergence in $L^1$: 1) $X_n\rightarrow X$ weakly in $L_1$ iff $\mathrm{E}(X_n\mathrm{1}_A)\rightarrow \mathrm{E}(X\mathrm{1}_A)$ for every measurable set $A$. 2) $X_n\rightarrow X$ weakly in $L_1$ iff $\mathrm{E}(X_n f)\rightarrow \mathrm{E}(X\mathrm{1}f)$ for every (essentially) bounded measurable function $f$. my question: are 1) and 2) equivalent? I see that 2) implies 1) (indicators […]

About Lusin's condition (N)

We say that $f:[0,1]\to \mathbb{R}$ satisfies Lusin’s condition (N) provided $$m(f(B))=0 \quad\mbox{whenever}\quad B\subseteq [0,1] \mbox{ with }m(B)=0$$ where $m$ stands for the Lebesgue measure on $\mathbb{R}$. I found in here the following definition. We say that $f:[0,1]\to X$ satisfies Lusin’s condition (N) provided $$\mathcal{H}^1(f(B))=0 \quad\mbox{whenever}\quad B\subseteq [0,1] \mbox{ with }m(B)=0$$ where $X$ is a metric […]

Is there a name for the “famous” inequality $1+x \leq e^x$?

Is there a name for the “famous” inequality $1+x \leq e^x$? It has many variants depending on how you arrange the terms: $$1 + x \leq e^x$$ $$e^{-x} -x – 1 \geq 0 $$ $$\ln(1+x) \leq x$$ Et cetera. Perhaps the simplest mnemonic device is, “$e^x$ lies above its tangent line at the origin.” This […]

A possible inequality related to binomial theorem (or, convex/concave functions)

Let $x, \ y, \ p$ be any real numbers with $x>0$, $y>0$, and $p>1$. The question is about (most probably) an elementary inequality: Is it always true that $x^p+y^p\leq (x+y)^p$ ? Note that if $p$ is any positive integer, then the above inequality is obviously correct. What about if the number $p \ (\text{with} […]

Proof of Newton's Binomial Expansion

I have some thoughts pertaining to the proof of Newton’s Binomial Expansion in Patrick Fritzpatrick’s Advanced Calculus. The theorem is stated as such: Let $\beta$ be any real number, then $$(1+ x)^\beta = \sum^{\infty}_{k=0}{{\beta} \choose {k}}x^k, |x| < 1.$$ Here is an excerpt of the proof: Consider the case where $-1 < x <0$. Write […]

Fundamental Theorem of Calculus for distributions.

Consider a function $F \in C^{\alpha}( \mathbb{R})$ for $0 < \alpha < 1.$ Then we can take it’s distributional derivative. We can say $f = F’ \in C^{\alpha -1}( \mathbb{R} )$. My issue is going back. Say I have a $\alpha -1 -$Hölder function $f$, then how can I “integrate” it to get a primitive […]

Lebesgue measure of any line in $\mathbb{R^2}$.

What is the Lebesgue measure of a line in $\mathbb R^2$? I am guessing that this zero. But i couldn’t prove it rigorously. Please help… From this can i conclude that any proper subspace of $\mathbb R^n$ has measure zero.

How to show that the monomials are not a Schauder basis for $C$

why the monomials are not a Schauder basis for $C[0,1]$? $p_n(x)=x^n$ such that $(p_n)$ does not form a Schauder basis for $C[0,1]$ span$\lbrace p_n : n\ge 0\rbrace$ is dense in $C[0,1]$ by Weierstrass approxmation theorem, but I cannot figure out why they are not a Schauder basis. Could you please explain

Extension of uniformly continuous function

I want to prove this: $f\in C((a,b))$ uniformly continuous. Then there exists $\tilde{f}\in C([a,b])$ extension of $f$. I took $x_n\rightarrow a$ and defined $\tilde{f}(a)=\mathrm{lim}\;f(x_n)$. I saw that this is a good definition, the only thing that I’m not able to prove is that $\tilde{f}$ is continuous at $a$ (or $b$). Could you help me please?