Let $f:[0,1] \to \mathbb{R}$ be a continuous function. Show that $\lim_{n \to \infty} \prod_{k=1}^{k=n} (1+ \frac{1}{n}f(\frac{k}{n})) = e^{\int_{0}^{1}f(x) dx}$. Attempt:  let $\lim \prod_{k=1}^{k=n} (1+ \frac{1}{n}f(\frac{k}{n})) = l$ then $\lim_{n \to \infty} \log(\prod_{k=1}^{k=n} (1+ \frac{1}{n}f(\frac{k}{n})))= \log l \implies \lim \sum_{k=1}^{k=n} \log(1+ \frac{1}{n}f(\frac{k}{n})) = \log l$. and working backwards we should have $\log l = \int_{0}^{1} f(x)dx = \lim_{n \to \infty} \sum_{k=1}^{k=n} \frac{1}{n} f(\frac{k}{n})$. So here I'm trying to find the link between $\lim \sum_{k=1}^{k=n} \log(1+ \frac{1}{n}f(\frac{k}{n}))$ and $\lim_{n \to \infty} \sum_{k=1}^{k=n} \frac{1}{n} f(\frac{k}{n})$. but $\log(1+x) \approx x$ for $x &lt;1$, so in order to use this approximation I have to prove that $\frac{1}{n}f(\frac{k}{n})&lt; 1$ (or maybe I can use the fact that f is continuous and prove that f attains supremum at some point in $[0,1]$) after some terms and I'm stuck .  Is there any other approach which avoids this approximation?
Given that $f:[0,1]\to \Bbb{R}$ is a continuous function, I need to show that $$\lim_{n \to \infty}\prod_{k=1}^{n}\left(1+\frac{1}{n}f\left(\frac{k}{n}\right)\right)=e^{\int_0^1f(x)dx}$$  Writing $$y=\lim_{n \to \infty}\prod_{k=1}^{n}\left(1+\frac{1}{n}f\left(\frac{k}{n}\right)\right)$$ and taking the logarithm of both sides and interchanging limit and logarithm(since $f$ is continuous), I get $$\log y = \lim_{n\to \infty}\sum_{k=1}^{n}\log\left(1+\frac{1}{n}f\left(\frac{k}{n}\right)\right)$$ Now I know that the limit of the Riemann sum $\lim_{n\to \infty}\sum_{k=1}^{n}\frac{b-a}{n}f\left(a+\frac{(b-a)}{n}k\right)=\int_a^bf(x)dx$. However if I want it to read $\int_0^1f(x)dx$ in the given problem, I would need the term inside the summation to read $\frac{1}{n}f\left(\frac{k}{n}\right)$. The way to do this it would seem is to take the series expansion of $\log(1+x)$ and ignoring the $x^2$ and higher order terms. Is it justifiable to do so?
How find this sum $\sum\limits_{n=0}^{\infty}\frac{1}{(3n+1)(3n+2)(3n+3)}$ Find this sum     $$I=\sum_{n=0}^{\infty}\dfrac{1}{(3n+1)(3n+2)(3n+3)}$$   My try: let     $$f(x)=\sum_{n=0}^{\infty}\dfrac{x^{3n+3}}{(3n+1)(3n+2)(3n+3)},|x|\le 1$$   then we have $$f^{(3)}(x)=\sum_{n=0}^{\infty}x^{3n}=\dfrac{1}{1-x^3}$$ then we find the $f(x)$,Following is very ugly(can you someone can post your follow solution,) have other simple methods? Thank you very much. StringTokenizer? Convert the String to a char[] and iterate over that? Something else? Is -1 followed by a singular or plural noun? Span class in opennlp is not working Does Earth's Rotation Affect Its Shape? The question I am working on is, &quot;Consider the following. (a) Find the angular speed of Earth's rotation about its axis. rad/s (b) How does this rotation affect the shape of Earth?&quot;  I am fully capable of solving part (a); however, I am not sure how to describe the effect earth's rotation on its shape. 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Suppose, I want to refer multiple equations (1), (2), (3), (4), (5) and (7) so that they appear as (1-5, 7):  \documentclass[fleqn,preprint,10pt]{article} \usepackage{amsmath} \usepackage{amssymb} \usepackage{amsfonts} \begin{document} \begin{align} \label{1}\psi([e_1,e_2]) = \psi(e_2) = e_2 = (e_2,2e_3) = (-2e_3,e_2) = (\psi(e_1), \psi(e_2))\\ \label{2}\psi([e_1,e_3]) = \psi(2e_3) = 2e_1 = (e_1,2e_3) = (-2e_3,e_1) = (\psi(e_1),\psi(e_3))\\ \label{3}\psi([e_2,e_3]) = 0 = (e_2,e_1) = (\psi(e_2),\psi(e_3)),\\ \label{4}\left\{e_4, e_1\right\},\,\,\,\left\{e_4, e_2\right\},\,\,\,\left\{e_4, e_3\right\},\,\,\,\left\{e_4, e_1+\epsilon\,e_2\right\},\\ \label{5}\left\{e_1, e_2\right\},\,\,\,\left\{e_3, e_1\right\},\,\,\,\left\{e_3, e_2\right\},\\ \label{6}\left\{e_4, e_1\right\},\,\,\,\left\{e_4, e_2\right\},\,\,\,\left\{e_4, e_3\right\},\,\,\,\left\{e_4, e_1+\epsilon\,e_2\right\},\\ \label{7}\text{Nor}_{L_4\oplus L_3}\left(\{e_2\}\right)/\{e_2\}=\,\{e_1, e_3, e_4\} \end{align} \end{document} How do I make sure that values from MySQL keep their type in PHP? Is there a way to add a timer to a Beamer presentation? I have seen presentations in Power Point which contain a timer. Can this be done in Beamer using a simple command, or maybe a package? If not a timer, is it possible to at least get a clock on your slides? I have been using Ubuntu 12.04 for quite a while and have never had a problem come at me out of the blue like this. I'm trying to install python 3.2 on my Ubuntu system and every time I run the code to compile the python source, I get this error:  'Command 'sudo' is available in '/usr/bin/sudo' The command could not be located because '/usr/bin' is not included in the PATH environment variable. sudo: command not found'   I've also tried running gksudo command to view what my environment shows but I pretty much get the same error. Focus existing terminal with `Ctrl-Alt-T` shortcut Why does a vertical line have a SLOPE (steepness) of undefined? The curve shows :   x goes 1 to right  y goes up by 3   the SLOPE is defined as the ratio of rise to the step  SLOPE = rise/step = 3/1 = 3  Now that :   Horizontal line has SLOPE of 0. Vertical line has SLOPE of undefined.   It makes sense for me to imagine horizontal line has SLOPE of 0 since there is no rise at all. But why the vertical has SLOPE of undefined.? Show that  $$\frac{\sin x}{\cos 3x}+\frac{\sin 3x}{\cos 9x}+\frac{\sin 9x}{\cos 27x} = \frac{1}{2}\left(\tan 27x-\tan x\right)$$ MySQL autoincrement column jumps by 10- why? I have a couple tables in which I created an object ID as either an Int or Bigint, and in both cases, they seem to autoincrement by 10 (ie, the first insert is object ID 1, the second is object ID 11, the third is object ID 21, etc).  Two questions:   Why does it do that? Is that a problem? declare property as object? How to detect if JavaScript is disabled? There was a post this morning asking about how many people disable JavaScript. 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So $\displaystyle \left(\frac{x\cdot 2014^{\frac{1}{x}}+\frac{1}{x}\cdot 2014^x}{2}\right)\geq \left(x\cdot 2014^{\frac{1}{x}}\cdot \frac{1}{x}\cdot 2014^x\right)^{\frac{1}{2}} = \sqrt{2014^{\left(x+\frac{1}{x}\right)}} = 2014$   And equality Hold , When $\displaystyle x = \frac{1}{x}\Rightarrow x= 1$  Can we solve it  without Using $\bf{A.M\geq G.M}$  If Yes, Then please explain me.  Thanks Is the C# "explicit implementation" of the interface present in Java? In C#, if you have two base interfaces with the same method (say, F()) you can use explicit implementation to perform different impl. for F(). This alloes you to differently treat the object, corresponding to the current point of view: as IMyInterface1 or IMyInterface2. Is this possible in Java? How do I access my Windows partition? I have Ubuntu 12.04 installed on my HP G62-453 TU Laptop. I cannot access the contents of my hard disk partition E.