Problem: $f'(x)$ is a continuous function for $[0, 1]$. Show that $$\lim_{n \to \infty}n \left( \frac{1}{n}\sum_{i=1}^n {f\left(\frac{i}{n}\right)-\int_0^1f(x)dx} \right)=\frac{f(1)-f(0)}{2}$$ I tried to use the definition of the definite integral to change it to a limit but it doesn’t seem to work. And I also wonder why a continuous condition for $f'(x)$ was given. I thought it […]

This question already has an answer here: Prove that $( 1 + n^{-2}) ^n \to 1$. 8 answers

$$\large f(x)= \lim_{n\rightarrow \infty}\left( \dfrac{n^n(x+n)\left( x+\dfrac{n}{2}\right)\left( x+\dfrac{n}{3}\right)… \left( x+\dfrac{n}{n}\right)}{n!(x^2+n^2)\left( x^2+\dfrac{n^2}{4}\right)\left( x^2+\dfrac{n^2}{9}\right)…\left( x^2+\dfrac{n^2}{n^2}\right)}\right)$$ $x\in R^+$ Find the coordinates of the maxima of $f(x)$. My Work: Is the method correct? Is there an easier way?

I’ve recently been going over the mean value and intermediate value theorems, however I’m not sure where to start on this.

Let $R$ be the region bounded by the graphs of $y = \cos \left(\frac{\pi x}{2}\right)$ and $y=x^2 -1$. The line $y=k$ splits the region $R$ into two equal parts. Find the value of $k$. First find the area. $$A = \int\limits_{-1}^1 \left[\cos \left(\frac{\pi x}{2}\right) – x^2 +1\right]\, \mathrm{d}x = \frac{4}{\pi} + \frac{4}{3}$$ Not exactly sure […]

The Harmonic number $H_n$ is defined as $H_n=\sum_{t=1}^n\frac{1}{t}$. I wish to compute $\lim_{n\to\infty}\frac{k}{n}(H_n-H_k)$ where $k$ is a function of $n$ (which can be a constant function, e.g. $k=3$). Moreover, I wish the computation to be as simple as possible and self-sufficient (which might already be too much to ask). If a general solution is not […]

$$x^y = y^x$$ is the equation My solution is :$$\frac { dy }{ dx } =\left( \ln { y-\frac { y }{ x } } \right) /\left( \ln { x-\frac { x }{ y } } \right) $$ Just wondering if this is correct!

I want to solve the following question. $$\lim_{(x,y)\rightarrow (0,0)}\dfrac{xy^2}{x^2-y^2}$$ I am going to use polar coordinates. $x=r\cos\theta$ and $y=r\sin\theta$ $$\lim_{(x,y)\rightarrow (0,0)}\dfrac{xy^2}{x^2-y^2}=\lim_{r\rightarrow 0^+}\frac{r^3\cos\theta\sin\theta}{r^2(\cos^2\theta-\sin^2\theta)}=\lim_{r\rightarrow 0^+}\dfrac{r\cos\theta\sin^2\theta}{\cos^2\theta-\sin^2\theta}=0$$ if $\theta\neq \dfrac{\pi}{4},\dfrac{3\pi}{4},\dfrac{5\pi}{4}, \dfrac{7\pi}{4}.$ For values of $\theta$ we have the lines $y=x$ and $y=-x$, but they are bot in the domain of the function. Wolfram says it does not exist. Why?

This question already has an answer here: Prove the series $ \sum_{n=1}^\infty \frac{1}{(n!)^2}$ converges to an irrational number 4 answers The sum of the series $\sum_{n=0}^{\infty}\frac{\epsilon_n}{n!}$ is an irrational number 2 answers

I am having trouble making sense of how to figure out what a graph looks like just by knowing the critical numbers and the intervals the function is increasing and when it is decreasing. For example I have $(-\infty, -1), (0,1)$ as negative and $(-1,0), (1,+\infty)$ as positive. I know that the it can only […]

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