Linked Questions
88 questions linked to/from When can a sum and integral be interchanged?
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$X_n$ are r.v.s, is it true that $E[\sum_{n=1}^{\infty} X_n] = \sum_{n=1}^{\infty} E[X_n] $? [duplicate]
$X_n$ are r.v.s, is it true that $E[\sum_{n=1}^{\infty} X_n] = \sum_{n=1}^{\infty} E[X_n] $?
My feeling is that this is not necessarily true. But cannot come up with an example.
Can someone provide ...
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Change of summation and integration [duplicate]
Let $0<\theta\leq 1$ and $a, b>0$. Then for $x>0$, we have $0< e^{-(ax+\frac{b}{2}x^2)}<1$. Making Binomial series expansion, we have
$$\left[1-e^{-(ax+\frac{b}{2}x^2)}\right]^{\theta-...
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If $X_n$ are independent random variables, then does $\sum_n \mathbb{E}(X_n)=\mathbb{E}(\sum_n X_n)$? [duplicate]
If $X_n$ are independent random variables, then does $\sum_n \mathbb{E}(X_n)=\mathbb{E}(\sum_n X_n)$?
This is not a homework problem but rather a question I had. If it is not true, what are the ...
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dumbed down explanation of Fubini's theorem? [duplicate]
Could someone please give me a simple explanation of when it is valid to interchange the order of Intergration and summation when the summation is an infante summation?
for some context during lecture ...
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The permutation of the signs of sum and integration [duplicate]
Is it possible to swap sum and integration signs in this integral
$$\int\limits_{0}^{+\infty}\sum\limits_{n=1}^{+\infty}a^nx^{1/2}e^{-nx}dx=\sum\limits_{n=1}^{+\infty}a^n\int\limits_{0}^{+\infty}x^{1/...
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Finding solutions to non elementary integrals [duplicate]
For integrands such as $e^{-x^{2}}$ is it possible to transform into a Taylor series and exchange the summation with the integral similar to the problem sophomore's dream? $$\int e^{-x^{2}}dx=\int \...
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Integrate: $ \int_0^\infty \frac{\log(x)}{(1+x^2)^2} \, dx $ without using complex analysis methods
Can this integral be solved without using any complex analysis methods: $$ \int_0^\infty \frac{\log(x)}{(1+x^2)^2} \, dx $$
Thanks.
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What is the value of the integral$\int_{0}^{+\infty} \frac{1-\cos t}{t} \, e^{-t} \, \mathrm{d}t$?
I have a question about evaluating
$$\int_{0}^{\infty} \frac{1-\cos t}{t} \, e^{-t} \, \mathrm{d} t$$
Since $\lim_{t \to 0} \frac{1-\cos(t)}{t} =0$, we know that the integrand is integrable near ...
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Reversing the Order of Integration and Summation
I am trying to understand when we can interchange the order of Integration and Summation. I am increasingly encountering Integrals; some of which are being solved by interchanging the order of ...
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Prove that $\int_0^\infty \frac{e^{\cos(ax)}\cos\left(\sin (ax)+bx\right)}{c^2+x^2}dx =\frac{\pi}{2c}\exp\left(e^{-ac}-bc\right)$
In my course, I have to prove formula below
$$I=\int_0^\infty \frac{e^{\cos(ax)}\cos\left(\sin (ax)+bx\right)}{c^2+x^2}dx =\frac{\pi}{2c}\exp\left(e^{-ac}-bc\right)$$
for $a,b,c>0.$
I know that ...
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Evaluating $\int_{0}^{1}\frac{x-1}{(x+1)\ln x} dx $ [duplicate]
Evaluate the following integral $$\displaystyle I=\int_{0}^{1}\frac{x-1}{(x+1)(\ln x)} \mathrm{d}x $$
My work: I tried it by letting $\displaystyle I(a)=\int_{0}^{1}\frac{(x-1)x^a}{(x+1)(\ln x)} \...
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Evaluate $ \int _{ 0 }^{ 1 }{ \ln\left(\frac { 1+x }{ 1-x } \right)\frac { dx }{ x\sqrt { 1-{ x }^{ 2 } } } }$
Problem: Evaluate:
$$\displaystyle I=\int _{ 0 }^{ 1 }{ \ln\bigg(\frac { 1+x }{ 1-x } \bigg)\frac { dx }{ x\sqrt { 1-{ x }^{ 2 } } } }$$
On Lucian Sir's advice, I substituted $x=\cos(\theta)$. ...
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4
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How to compute $\int_0^{\frac{\pi}{2}} \frac{\ln(\sin(x))}{\cot(x)}\ \text{d}x$ [closed]
I am trying to compute this integral.
$$\int_0^{\frac{\pi}{2}} \frac{\ln(\sin(x))}{\cot(x)}\ \text{d}x$$
Any thoughts will help. Thanks.
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Is the integral of the sum really the sum of the integrals?
I was asked to find the mclaurin series of $\int_0^x\frac{\arctan (t)}{t}dt$
using the known mclaurin for arctan: $\arctan(t)=\sum_{n=1}^{\infty} \frac{(-1)^{n+1}t^{2n-1}}{2n-1}$
Ok, so what I did ...
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Evaluate $\int_0^{\infty} x^2\ln(\sinh x)\operatorname{sech}(3 x){\rm d}x $.
Problem
Evaluate $\displaystyle\int_0^{\infty} x^2\ln(\sinh x)\operatorname{sech}(3 x){\rm
d}x .$
Someone writes as follows
\begin{align*}
&\int_0^{\infty} x^2\ln(\sinh x)\operatorname{sech}(3 ...
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