# Tagged Questions

Mathematical analysis. Consider a more specific tag instead: (real-analysis), (complex-analysis), (functional-analysis), (fourier-analysis), (measure-theory), (calculus-of-variations), etc. For data analysis, use (data-analysis).

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### Multiplication of infinite series

Why multiplication of finite sums $(\sum_{i=0}^n a_i)(\sum_{i=0}^n b_i)=\sum_{i=0}^n (\sum_{j=0}^ia_jb_{i-j})$ (EDIT: This assumption was shown to be false) does not work in infinite case? I have ...
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### What is the convex-hull of the set $\{ (n,\varphi(n)) : n\in \mathbb N \} \subset \mathbb R^2$

I know that set $$E=\{ (n,\varphi(n)) : n\in \mathbb N \} \subset \mathbb R^2$$ has infinitely many points on the line $y=x-1$, which suggests this line to be included in the upper part of the ...
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### Open and connected in $R^n$ revised

I am trying to understand the following: If we have an open and connected set in $R^n$ then it can be connected with line segments parallel to the axes. I managed to prove this: If a set $U$ is open ...
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### Showing a Cauchy sequence does not have a limit

Okay basically I've managed to work through parts 1 a and b (with some help) now I'm a little stuck on part c). I think I can show fn is a cauchy sequence by virtue of the fact that fn-f tends to 0, ...
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### $\lim \sup$ of a sequence

Let $\{A_n\}$ be a sequence and $\frac{1}{R} = \lim \sup A_n$. Let $\alpha < R$. My question: Why is there $n_0\in \Bbb N$ such that $$A_n < \frac{1}{\alpha}\text{ for any } n\geq n_0$$ Thanks ...
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### Use L'Hopital's Rule to Prove

Let $$f: \mathbb R\rightarrow \mathbb R$$ be differentiable, let a in $\mathbb R$. Suppose that $f''(a)$ exists. Prove that $$\lim_{h\rightarrow0}\frac{f(a+h)-2f(a)+f(a-h)}{h^2}=f''(a)$$ Suppose ...
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### Showing the radius of convergence of $\sum a_n b_n z^n$ is at least $R_1 R_2$
Problem: If $\sum a_n z^n$ and $\sum b_n z^n$ have radii of convergence $R_1$ and $R_2$, show that the radius of convergence of $\sum a_n b_n z^n$ is at least $R_1 R_2$. Is the following proof ...