Theoretical foundations of calculus: limits, convergence of sequences, construction of the real numbers, least upper bound property, and related analysis topics such as continuity, differentiation, integration through the fundamental theorem of calculus.

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Convergence of Euler-transformed zeta series

I am trying to prove that the expression $$(1-2^{1-s})\zeta(s)=\sum_{n=0}^\infty\sum_{m=0}^n\frac{(-1)^m}{2^{n+1}}{n\choose m}(m+1)^{-s}$$ converges to an analytic continuation of the alternating ...
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1answer
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Question on differentiate under integral

First we have the following theorem: Then we apply it to a concrete problem: Finally how to obtain the second rectangle?
3
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7answers
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Assume that $ f ∈ L([a, b])$ and $\int x^nf(x)dx=0$ for $n=0,1,2…$.

Assume that $ f ∈ L([a, b])$ and $\int x^nf(x)dx=0$ for $n=0,1,2...$. Prove $f=0$ a.e. Since there exist polynomials going to f almost everywhere all I would need to do is bring the limit in to ...
3
votes
0answers
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What types of integrals cannot be solved using improper Riemann-Stieltjes Integration?

I came across the wikipedia discussion of the Riemann-Stieltjes integral. The first sentence in the "Generalization" section gave me pause: An important generalization is the Lebesgue–Stieltjes ...
0
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0answers
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quasi-convexity of a function

Can someone help me identify whether the following function is quasi-convex? Let $p>1$. For $x=(x_1,\dots,x_n),x_i>0,\sum_ix_i=1$, we define $$f(x) = -\log \sum_i (x_i/\|x\|_p)^{p-1}.$$ Plots ...
4
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Functions with compact support

I have a question about a convergence of functions with compact support. SETTING Let $d\geq 3$ and $U \subset \mathbb{R^{d}}$ be open and $dx$= Lebesgue measure on $U$. Let $b_{i},c,d_{i} \in ...
3
votes
0answers
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norm of a singular integral operator

My question is from Harmonic Analysis, about the study of singular kernels (in the Calderon Zygmund sense.) Suppose that a kernel $K$ is a singular kernel, extending to a bounded operator on $L^p$, ...
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1answer
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find a $B_{n,j}$ such that $|A_{n,j}-L_j| \leq B_{n,j}$ $\forall n,j$ and $\sum_{j=0}^{\infty}B_{n,j}$ converges

We have $A_{n,j}= 3(-1)^j2^{n-j+1}\frac{(2(n-j)-4)!}{(n-j)!(n-j-2)!}\binom{j+2}{2}\frac{n^\frac{5}{2}}{8^n}$ and $L_j=(-\frac{1}{8})^j\binom{j+2}{2}\frac{3}{8\sqrt{\pi}}$ So I know $\lim_{n \to ...
0
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1answer
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Prove that the functions $g_k(z) = f_k \circ h_k(z)$ form a normal family.

I am having a bit of trouble with the following complex analysis question which originates from a qual. Some help would be awesome. Let $f_k :\mathbb{D} \rightarrow \mathbb{C}$ be a normal family of ...