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interchanging
▪ I. interˈchanging, vbl. n. [f. as prec. + -ing1.] The action of the vb. interchange, in various senses; mutual or alternate exchanging.c 1374 Chaucer Boeth. i. met. v. 14 (Camb. MS.) Whi suffres thow þat slydynge fortune torneth so grete entrechaunginges of thinges? Ibid. iv. met. iv. 102 They moe... Oxford English Dictionary
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Interchanging sum and Riemann Integral The question: Is there a theorem that allows you to interchange sum and integral for positive functions sequences, but the theorem and it's proof only involves riemann or riemann...
I suggest you have a look at < It is a quite good clarification.
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Why for a multilinear form $w(X,Y,Z)$ it suffices to say that interchanging $X$ and $Y$ and $X$ and $Z$ changes the sign for $w$ to be alternating? Why for a multilinear form $w(X,Y,Z)$ it suffices to say that interch...
If you have a multilinear form $w$ with $n$-variables $(x_1,x_2\dots,x_n)$, you have to check that, for every $\sigma \in S_n$ ($S_n$ being the symmetric group) : $$w(x_{\sigma(1)},x_{\sigma(2)}\dots,x_{\sigma(n)}))=sign(\sigma)w(x_1,x_2\dots,x_n).$$ With $sign(\cdot)$ being the signature of the per...
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interchanging integrals Why does $$\int_0^{y/2} \int_0^\infty e^{x-y} \ dy \ dx \neq \int_0^\infty \int_0^{y/2} e^{x-y} \ dx \ dy$$ The RHS is 1 and the LHS side is not. Would this still be a legitimate joint pdf eve...
$$\int_0^{\infty} \int_0^{y/2} \exp(x-y) dx dy = \int_0^{\infty} \int_{2x}^{\infty} \exp(x-y) dy dx$$ Note that both, not surprisingly, yield the same answer. $$\int_0^{\infty} \int_0^{y/2} \exp(x-y) dx dy = \int_0^{\infty} (\exp(-y/2) - \exp(-y)) dy = 1$$ $$\int_0^{\infty} \int_{2x}^{\infty} \exp(x...
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Interchanging the order of a double infinite sum I'm stuck at a proof of Wald's first equation about interchanging the order of a double infinite sum: > Suppose $X_n \ge 0$ and $1_{\\{\cdot \\}}$ be indicator functi...
Note that $$\sum_{k\geqslant 1}\sum_{n\leqslant k}=\sum_{k\geqslant 1}\sum_{n\geqslant 1}[n\leqslant k]=\sum_{n\geqslant 1}\sum_{k\geqslant 1}[n\leqslant k]=\sum_{n\geqslant 1}\sum_{k\geqslant 1}[k\geqslant n]=\sum_{n\geqslant 1}\sum_{k\geqslant n}$$ This is an example of the usefulness of the Ivers...
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Expected value of a continuous random variable: interchanging the order of integration I have come across a proof of the following in Ross's book on Probability - For a non-negative continuous random variable Y with ...
we have \begin{align*} \int_{[0,\infty)}\int_{[y,\infty)} f_Y(x)\; dx\, dy &= \int_{[0,\infty)}\int_{[0,\infty)}\chi_{[y,\infty)}(x)f_Y(x)\;dx\,dy\\\ &= \int_{[0,\infty)}\int_{[0,\infty)}\chi_{[y,\infty)}(x)f_Y(x)\;dy\,dx\\\ &= \int_{[0,\infty)}\int_{[0,\infty)}\chi_{[y,\infty)}(x)\;dy\cdot f_Y(x)\;...
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Interchanging order of limits How do we reconcile the following difference? Specifically, what prevents us from changing the order of limits here? $$\lim_{\lambda \to \infty} \sum_{n=0}^\infty e^{-\lambda} \lambda^n/...
The polynomial dose not converges to $e^{\lambda}$ uniformly on the whole real line, so you cannot carelessly interchanging the limit and summation.
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Interchanging expecation and a monotone function Are there any theorems similar to Jensen's Inequality for monotone functions instead of convex functions? For example, if I know $EX \le EY$, can I say anything about $...
If $X$ and $Y$ are stochastically ordered, i.e. say $X\leq_{st.}Y$, then we have both $$\mathbb{E}(X)\leq \mathbb{E}Y\ \mbox{and}\ \mathbb{E}(f(X))\leq \mathbb{E}(f(Y))$$ where $f$ is a non-decreasing function. Conversely, if for any non-decreasing function $f$ we have $$\mathbb{E}(f(X))\leq \mathbb...
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Interchanging limits and logarithms This is probably not too smart, just wondering of the name of this rule: $$ \log \lim_{x \to x_0}f(x) = \lim_{x \to x_0}\log f(x) $$ A reference to a source and/or proof would be go...
Search for "interchanging limits," and you're sure to find sources. Edit: Here's something worthwhile from an introductory real analysis textbook.
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How can I solve the following series? I've thought about interchanging the order of summation but I don't know how and what's its rules I also have problem in solving the infinite series. !series
You can interchange the order of summation provided all the terms are nonnegative; in this case they are (this is a special case of Tonelli's theorem). This gives **Your sum** $ = \sum_{j=1}^n \sum_{i=0}^{\infty} (\dfrac{j}{j+1})^i = \sum_{j=1}^n \dfrac{1}{1-\dfrac{j}{j+2}} = \frac{1}{2} \sum_{j=1}^...
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Simple Integration Identity > Prove, by sketching the region of integration and interchanging the order of integration, that > > $$ \int_a^x \int_a^{\xi} f(s) ds d\xi = \int_a^x (x-s)f(s)ds $$ This is probably quite...
Note that the integration region is a triangular region with vertices at $(a,a)$, $(a,x)$, and $(x,x)$ in the $s-\xi$-plane. Thus, if the inner integral is on $s$, we see that for any fixed $\xi$, $s$ begins at $a$ and ends at $\xi$. If the inner integral is on $\xi$, we see that for any fixed $s$, ...
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Is there something wrong with my interchanging of sums and integrals? > Is there something wrong with my interchanging of sums and integrals? Consider $$\sum_{n\leq x}a_n(x-n)=\int_1^xA(t)dt,$$ where $A(x)=\sum_{n\le...
The error is in the integral. When you use partial summation, you work with integers. So, in your case$$\int_{1}^{x}A\left(t\right)dt\neq\int_{1}^{x}\left(t+1\right)tdt$$ but, since you're working now in an continuous context,$$\int_{1}^{x}A\left(t\right)dt=\int_{1}^{x}\left(\left\lfloor t\right\rfl...
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limit of integral and uniform convergence Assume that $f_n : [0,1] \to \mathbb{R}$ are integrable and that $(f_n)$ uniformly converges to $f$. Does the limit $\lim \limits_{n \to \infty} \int \limits_{0}^{1} f_n(x)^{2...
$g(x)$ has a bounded derivative on $[0,1]$ so by the MVT $g(x)$ is Lipschitz-continuous on $[0,1.$ So $g(f_n)$ converges uniformly to $g(f)$ on $[0,1$. And each $g(f_n)$ is integrable so $g(f)$ is integrable. Hence $$\lim_{n\to \infty}\int_0^1g(f_n(x))dx=\int_0^1g(f(x)dx=\int_0^1\lim_{n\to \infty}g(...
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Interchanging limits without Fubini/Tonelli? I'm working on a problem and have reached the following $$ \int_{0}^{1}\sum_{n=0}^{\infty}\frac{(x\log x)^{n}}{n!}\;dx $$ and I'd like to interchange the integral and the...
**Hint** : prove that the series $\sum_{n=0}^{\infty}\frac{(x\log x)^{n}}{n!}$ converges uniformly in $[0,1]$.
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The principle of duality for sets The Wikipedia article on the algebra of sets briefly mentions the following: > These are examples of an extremely important and powerful property of set algebra, namely, the principl...
As written,the statement is false. For example, the statement $\neg (\exists x)(x\in \varnothing)$ won't flip around that way. The actual point, however, is that the set of all subsets of a set $U$ forms a lattice with join $\cup$, meet $\cap$, bottom $\varnothing$, top $U$, and ordering $\subseteq$...
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