# Tagged Questions

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### Mass of function concentrating near origin

Let $$g_t(x)=\dfrac{1}{\sqrt{t}}e^{\frac{-(at+x)^2}{4t}}$$ where $a\in\mathbb{R}$ and $t>0$. Fix $r>0$. Why is it true that $$\lim_{t\rightarrow 0^+}\int_{|y|>r}g_t(y)dy=0?$$ We can show that ...
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### Integral of convolution difference approaches zero

Let $u(x,t)=f(x)\ast\left(\dfrac{1}{2\sqrt{\pi t}}e^{-\dfrac{(at+x)^2}{4t}}\right)$, and suppose that $f\in L^1$. Show that $$\lim_{t\rightarrow 0^+}\int_{-\infty}^\infty|u(x,t)-f(x)|dx=0$$ How ...
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### Integral of $y(x)$ when $y(t)$ is in the equation

I'm supposed to find the limit of $y(x)$ when $x \rightarrow \infty$ if $y$ is given by: $$y(x)=7+\int_0^x 4\frac{(y(t))^{2}}{1+t^2}dt$$ What I don't get is the $y(t)$ inside the integral. If I ...
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### Prove that $\lim_{b\to\infty}_{a\to0+}\int_a^b\frac{\hat{f}(\xi)}\xi d\xi=-\pi i\int_0^\infty f(x)dx$

Suppose $f\in L^1(\mathbb{R})$ and that $f$ is odd. Prove that $$\lim_{b\to\infty}_{a\to0+}\int_a^b\frac{\hat{f}(\xi)}\xi d\xi=-\pi i\int_0^\infty f(x)dx$$ Here $\hat{f}$ denotes the Fourier transform ...
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### Is there ever a requirement to change the limits of integration?

I don't have issues with doing integration problems, but occasionally I see the solution changing the limits of integration whenever a $u$-substitution is done. I obviously don't have a problem doing ...
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### Integral from $0$ to $\infty$ of $\frac{1}{3}\ln\left(\frac{x+1}{\sqrt{x^2-x+1}}\right)+\frac{1}{\sqrt 3}\arctan \left(\frac{2x-1}{\sqrt 3} \right)$

Evaluate the integral $$\int_0^\infty \left( \frac{1}{3}\ln\left(\frac{x+1}{\sqrt{x^2-x+1}}\right)+\frac{1}{\sqrt 3}\arctan \left(\frac{2x-1}{\sqrt 3} \right) \right) dx$$ I have read about ...
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### What am I doing wrong? Integration and limits

I need some help identifying what i'm doing wrong here.. What is the limit of y(x) when x→∞ if y is given by: $$y(x) = 10 + \int_0^x \frac{22(y(t))^2}{1 + t^2}\,dt$$ What i've done: 1) Integrating ...
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### Show that $\lim_{n\to\infty}\sum_{k=1}^{n}\frac{n}{n^2+k^2}= \frac{\pi}{4}$ [duplicate]

Show that: $$\lim_{n\to\infty}\sum_{k=1}^{n}\frac{n}{n^2+k^2}= \frac{\pi}{4}$$ My idea: I thought that this could be rewritten as an integral, then use trig substitution (perhaps tangent) and then ...
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### Counter-example for interchanging integral and limit

Consider $f_n \ge 0$ and $f_n \downarrow f$, can you provide a counter example (if it's not true) for $$\int f_n d\mu \downarrow \int f d\mu$$ And if the statement above is true, how can I prove ...
### Finding $\lim_{n\to\infty} e^{-n}\sum_{k=0}^n \frac{n^k}{k!}$ if it exists [duplicate]
Does there exist the following limitation? If the answer is yes, could you show me how to find that? $$\lim_{n\to\infty} e^{-n}\sum_{k=0}^n \frac{n^k}{k!}$$ In the following, I'm going to write what ...
I am trying to find the limit of the following integral as $n$ grows to infinity. $$\int_1^{ n+1 }\frac { \ln x }{ x^n } \, dx$$ However, when I am doing an integration by parts, the result looks ...