a limit about exponential function $\lim_{n\rightarrow\infty}\frac{1+\frac{n}{1!}+\cdot+\frac{n^n}{n!}}{e^n}=\frac12$  Taking the first $n$ terms of the Taylor series of $e^n$ as the numerator, the limit is true or false? How to prove?
Evaluating $\lim\limits_{n\to\infty} e^{-n} \sum\limits_{k=0}^{n} \frac{n^k}{k!}$ I'm supposed to calculate: $$\lim_{n\to\infty} e^{-n} \sum_{k=0}^{n} \frac{n^k}{k!}$$ By using WolframAlpha, I might guess that the limit is $\frac{1}{2}$, which is a pretty interesting and nice result. I wonder in which ways we may approach it.
Regression for an outcome (ratio or fraction) between 0 and 1 I am thinking of building a model predicting a ratio $a/b$, where $a \le b$ and $a &gt; 0$ and $b &gt; 0$. So, the ratio would be between $0$ and $1$.  I could use linear regression, although it doesn't naturally limit to 0..1.  I have no reason to believe the relationship is linear, but of course it is often used anyway, as a simple first model.  I could use a logistic regression, although it is normally used to predict the probability of a two-state outcome, not to predict a continuous value from the range 0..1.  Knowing nothing more, would you use linear regression, logistic regression, or hidden option c? Minimize absolute values of errors instead of squares How can I take shallow depth of field photos with a point-and-shoot camera? I have a compact digital camera, and in macro mode I can manage to achieve a blurred background, if the background is far enough away. Can this be done when taking non-macro pictures, and if not, is there a way to simulate it?  When I shoot portraits with my brother's DSLR, it is quite easy to fade out the background. But with my point and shoot camera, I haven't been able to take such photos.   I choose a large aperture for my camera but it still doesn't work. I know that standing further and zooming will result in a smaller depth of field, but it doesn't seem to be enough to get the really nice "bokeh" look. I've tried macro mode, and that will work if the subject is close enough and the background far enough away, but how can it be done with non-macro pictures?   Why are DSLR cameras are so much better in this area? What are the general recommendations to do this using a compact or super-zoom camera? dd aplied to dev USB before connect corrupt /dev/sdX and dont work Opposite of Alternatives The calling thread cannot access this object because a different thread owns it Question is :     Prove that the polynomial $p(x)=x^4-4x^2+8x+2$ is irreducible over the quadratic field $F=\mathbb{Q}(\sqrt{-2})$. [Hint : first use the method of proposition $11$ for the U.F.D $\mathbb{Z}[\sqrt{-2}]$(cf.Exercise $8$, Section $8.1$) to show that if $\alpha \in \mathbb{Z}[\sqrt{-2}]$ is a root of $p(x)$ then $\alpha$ is a divisor of $2$ in $\mathbb{Z}[\sqrt{-2}]$ . Conclude that $\alpha$ must be $\pm 1,\pm \sqrt{-2}$ or $\pm 2$ and hence show that $p(x)$ has no linear factor over $F$. Show similarly that $p(x)$ is not the product of quadratics with coefficients in $F$.]   What I have done so far is :  Suppose $\alpha \in\mathbb{Z}[\sqrt{-2}]$ is a root of $p(x)=x^4-4x^2+8x+2$ we would then have :  $p(\alpha)=\alpha^4-4\alpha^2+8\alpha+2=0\Rightarrow 2=\alpha(-\alpha^3+4\alpha-8)$  i.e., $\alpha$ is a divisor of $2$ in $\mathbb{Z}[\sqrt{-2}]$.  so, I have used the method of proposition $11$ for the U.F.D $\mathbb{Z}[\sqrt{-2}]$(cf.Exercise $8$, Section $8.1$) to show that if $\alpha \in \mathbb{Z}[\sqrt{-2}]$ is a root of $p(x)$ then $\alpha$ is a divisor of $2$ in $\mathbb{Z}[\sqrt{-2}]$  I do not understand why does he mentioned that $\mathbb{Z}[\sqrt{-2}]$ is U.F.D and all... I do not use that at all... It is unnecessarily confusing me or i am unnecessarily getting confused.. hint is actually misleading me :(  Now, I have to prove that $\alpha$ must be $\pm 1,\pm \sqrt{-2}$ or $\pm 2$  i.e., suppose I have $2=ab$ in $\mathbb{Z}[\sqrt{-2}]$ then, $N(2)=N(ab)\Rightarrow 4=N(a)N(b)$   i.e.,$N(a)=1\text{ or }2\text{ or } 4$ i.e., $p^2+2q^2=1\text{ or }2\text{ or } 4$ for $a=p+\sqrt{-2}q$   $p^2+2q^2=1\Rightarrow p=\pm 1 \Rightarrow a=\pm 1$ $p^2+2q^2=2\Rightarrow q=\pm 1\Rightarrow a=\pm\sqrt{-2}$  $p^2+2q^2=4\Rightarrow p=\pm 2\Rightarrow a=\pm 2$    Once I prove that those are the only divisors then I would consider :    $p(1)=(1)^4-4(1)^2+8(1)+2\neq 0$ $p(-1)=(-1)^4-4(-1)^2+8(-1)+2\neq 0$ $p(\sqrt{-2})=(\sqrt{-2})^4-4(\sqrt{-2})^2+8(\sqrt{-2})+2\neq 0$ $p(-\sqrt{-2})=(-\sqrt{-2})^4-4(-\sqrt{-2})^2+8(-\sqrt{-2})+2\neq 0$ $p(2)=(2)^4-4(2)^2+8(2)+2\neq 0$ $p(-2)=(-2)^4-4(-2)^2+8(-2)+2\neq 0$   So, no divisor of $2$ is a root..  Thus $p(x)$ do not have a root in $\mathbb{Z}[\sqrt{-2}]$  suppose I have something like :  $$x^4-4x^2+8x+2=(x^2+ax+b)(x^2+cx+d)=x^4+(a+c)x^3+(b+d+ac)x^2+(ad+bc)x+bd$$  But then, $bd=2\Rightarrow \text { b,d  are  a divisors of 2 in $\mathbb{Z}\sqrt{-2}$}$  But then we have seen that only divisors of $2$ in $\mathbb{Z}\sqrt{-2}$ are $\pm 1,\pm \sqrt{-2}$ or $\pm 2$  So, only possibilities are   $x^4-4x^2+8x+2=(x^2+ax\pm 1)(x^2+cx\mp 2)$ $x^4-4x^2+8x+2=(x^2+ax\pm \sqrt{-2})(x^2+cx\mp \sqrt{-2})$   But these are not possible...  This only tells that $p(x)$ is irreducible in $\mathbb{Z}[\sqrt{-2}]$ but then how do i show this is irreducible in $\mathbb{Q}(\sqrt{-2})$ I was expecting gauss lemma to help but it only works for integers and rationals...  So, please help me to clear this..  Thank you...   P.S : Proposition $11$ is Rational root theorem and Exercise $8$ is that ring of integers of $\mathbb{Q}(\sqrt{-2})$ is an Euclidean domain. Round function in Excel, worksheet function vs VBA 'sudo su -' vs 'sudo -i' vs 'sudo /bin/bash' - when does it matter which is used, or does it matter at all? Remove specific duplicates (all but latest) When removing duplicate rows, using the below query from  tutorial, how would I go about forcing which of the found duplicates to remove?  DELETE FROM dbo.ATTENDANCE      WHERE AUTOID NOT IN (SELECT MIN(AUTOID) _ FROM dbo.ATTENDANCE      GROUP BY EMPLOYEE_ID,ATTENDANCE_DATE)    And this works great. The reason why I am using this one is because the only unique id available is one in the IDENTITY column. And to determine if there are duplicate rows I have to look at a combination of multiple columns.    But if I have a set of duplicate rows how do I / how does SQL Server decide which to remove? And how would I force it to remove all but the one with the highest IDENTITY value?  EMPLOYEE_ID     ATTENDANCE_DATE     AUTOID A001            2011-01-01          1 A001            2011-01-01          2   If I would run the query now it happens to remove the second one, with AUTOID 2. But I am trying to remove all but this one (because this is the one latest added). Router to router setup works fine with straight cable up to 2-3 meter it is not working for 45 meter why? Similar matrices and field extensions Prove that to each $\epsilon >0$, there exists a $\delta >0$ so that the Lebesgue integral... Suppose $f$ is in $L^1$ space of $\mu$, where $\mu$ is the Lebesgue measure. Prove that to each $\epsilon &gt;0$, there exists a $\delta &gt;0$ so that the Lebesgue integral of the absolute value of $f$ is less than $\epsilon$ (over the set $E$) whenever the measure of $E$ is less than $\delta$.  I know that the integral of $f$ is finite. So for arbitrary $f$ we can find the integral of $f$ to be less than some $\epsilon$, but how would I connect that with the measure of $E$ less than $\delta$. Any help would be welcome. Why do only two sexes exist for animals? Why, from the natural selection point of view, do only two sexes exist for animals? Remove subscribers from Auto Suppression List Can I argue that:  \begin{align} \left(\int_a^b f(x) \,dx\right) \left(\int_c^d g(y) \,dy\right) &amp;= \int_a^b f(x) \int_c^d g(y) \,dy \,dx \\ &amp;= \int_a^b \int_c^d f(x) g(y) \,dy \,dx \end{align}  since the same would be true for a product of finite sums? Homeomorphism(topological spaces) version of Cantor–Bernstein–Schroeder theorem Word for someone who pays attention to details Stateful bash function Consider a text file with the following entries:  aaa bbb ccc ddd eee fff ggg hhh iii   Given a pattern (e.g. fff), I would like to grep the file above to get in the output:  all_lines except (pattern_matching_lines  U (B lines_before) U (A lines_after))   For example, if B = 2 and A = 1, the output with pattern = fff should be:  aaa bbb ccc hhh iii   How can I do this with grep or other command line tools?    Note, when I try:  grep -v 'fff'  -A1 -B2 file.txt   I don't get what I want. I instead get:  aaa bbb ccc ddd eee fff -- -- fff ggg hhh iii