Questions tagged [real-analysis]
For questions about real analysis, such as limits, convergence of sequences, properties of the real numbers, the least upper bound property, and related analysis topics such as continuity, differentiation, and integration.
146,895
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Is this step in the asymptotic analysis of a certain physical system ok?
In analyzing the thermodynamic limit of a certain system in statistical mechanics, I've encountered the following situation: $f_n$ is a sequence of probability density functions on the real line with ...
- 1,851
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Bound on the expectation of the maximum of a sequence, given bounds on the expected value of each element.
I have a sequence of independent random variables $U_1, U_2, \dots,U_N$.
Suppose $\mathbb{E}[U_i] \leq 1$ for all $i=1,\dots,N$, and let:
$$M_N = \max_{i=1,\dots,N} U_i$$
It is easy to see that $\...
- 21
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Does there exist a positive sequence with these properties?
Let $\{x\}$ denote the fractional part of $x$ and $d_n=\text{lcm}(1,2,...,n)$. Does there exist a sequence with all positive terms $(a_n)_{n\geq 1}$ such that $$\lim_{n\to\infty} \{a_n\}=1\ \text{,}\ \...
- 926
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Gradient and intermediate value theorem
I'm having trouble with an optimization problem. I have a continuous and concave objective function $\mathcal{O} : \mathbb{R}^n \longrightarrow \mathbb{R}$. I have strong evidence (numerical ...
- 31
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Convergence of the exponentially-weighted moving average in integral form
Let $f:[0,+\infty)\to\mathbb{R}$ be continuous and let $\bar{f}\in\mathbb{R}$ be such that $f(t)\to\bar{f}$ as $t\to+\infty$. Does this suffice for
\begin{equation}
\int_{0}^{t}{\rm e}^{-\rho(t-s)}f(...
- 301
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73
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Does $\int_a^b g^2=0 $ imply $\int_a^b fg=0$?
I am trying to prove the Cauchy Schwarz inequality for integrable functions, which is as follows:
(Cauchy-Schwarz): If $f$ and $g$ are integrable on $[a,b]$, then $$\left(\int_{a}^{b}f^{2}\right)\...
- 577
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Sobolev functions approximated by ridge functions
Let $f \in W^{k,2}(\mathbb{R}^d)$, a Sobolev space with smoothness $k$ and dimension $d$. We aim to approximate
$f$ using ridge functions of the form $g(\mathbf{a}.\mathbf{x})$. Suppose the ...
- 721
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If $\sum_{n=m}^\infty a_n$ diverges, does $\sum_{n=m+k}^\infty a_n$ diverge too? And the converse? [closed]
In Analysis I, Terrance Tao, third edition, there's the following result:
Let $\sum_{n=m}^\infty a_n$ be a series of real numbers, and let $k \geq 0$ be an integer. If one of the two series $\sum_{n=...
- 1,454
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Baby Rudin: Corollary to Theorem 2.12 (at most countable union of at most countable sets is countable)
I am trying to prove the corollary to Theorem 2.12 in Rudin's book. Theorem 2.12 says:
Let $\{E_n\}$, $n = 1,2,3, \ldots$ be a sequence of countable sets, and put $S = \bigcup\limits_{n=1}^{\infty} ...
- 1,281
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Does this slight modification alter the limit of this sequence?
Let $x\in\mathbf R$. Is the following true?
\begin{align}
\lim_{n\to\infty} \left[1- \frac{x^2}{n}\left(1+\frac{2}{3}\frac{x}{\sqrt{3n}}\right)\right]^{n/2} = \lim_{n\to\infty} \left[1- \frac{x^2}{...
- 1,851
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71
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Number of real roots of a specific polynomial
I am working with the following polynomial:
$f_n := (x^2 + 2(n − 1)) \cdot (x − 2) \cdot (x − 4) \cdot (x-6) \cdots (x − 2(n − 2)) − 2$; $n\ge3$
I had to show that $f_n$ has exactly $n-2$ real roots.
...
- 131
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2
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Discretized Distributions on Rationals?
Consider the measure space $(\mathbb{Q}, 2^{\mathbb{Q}}, \nu)$, with $\nu$ being the counting measure. The space is then $\sigma$-finite. Is there any attempts made to define analogues of continuous ...
- 2,497
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Closed form for the area under $ f(x):=\lim_{N \to \infty}\frac{\pi(Nx)}{\pi(N)} $
Define a function $f:\Bbb Q \to \Bbb Q$ by the following $$ f(x):=\lim_{N \to \infty}\frac{\pi(Nx)}{\pi(N)} $$
where $\pi(\cdot)$ is the prime counting function and $N\in \Bbb N.$ I would like to find ...
- 754
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3
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Real Analysis Question about Limit points and ε-neighborhoods
The question says "Prove that a point $x$ is a limit point of a set $A$ iff every ε-neighborhood of $x$ intersects $A$ at some point other than $x$."
I am having trouble proving the reverse ...
- 81
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A limit involving the $n$th derivative of the reciprocal of the Riemann Zeta function: $\lim _{n \to \infty} |(1/\zeta)^{(n)}(2)|$ [closed]
Does the limit $\lim _{n \to \infty} |(1/\zeta)^{(n)}(2)|$ exist? If so, how can we determine its precise value, or is there any way to approximate it with high precision? ($(1/\zeta)^{(n)}(2)$ is ...
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