Articles of limits

Riemann sum on infinite interval

It is well known that in the case of a finite interval $[0,1]$ with a partition of equal size $1/n$, we have: $$\lim_{n\rightarrow \infty} \frac{1}{n}\sum_{k=0}^{n-1} f\left(\frac{k}{n}\right)=\int_0^1 f(x)dx$$ I was wondering under which conditions on $f$ this could be extended to the case of the positive real line, i.e which conditions on $f$ would enable us […]

Evaluation of $\lim\limits_{n\to\infty} (\sqrt{n^2 + n} – \sqrt{n^3 + n^2}) $

Could you, please, check if I solved it right. \begin{align*} \lim_{n \rightarrow \infty} (\sqrt{n^2 + n} – \sqrt[3]{n^3 + n^2}) &= \lim_{n \rightarrow \infty} \sqrt{n^2(1 + \frac1n)} – \sqrt[3]{n^3(1 + \frac1n)})\\ &= \lim_{n \rightarrow \infty} (\sqrt{n^2} – \sqrt[3]{n^3})\\ &= \lim_{n \rightarrow \infty} (n – n) = 0. \end{align*}

Is this a valid proof of $\lim _{n\rightarrow \infty }(1+\frac{z}{n})^n=e^z$?

Define the function $g_n\left(z\right)=\left(1+\frac{z}{n}\right)^n$ for $\:n\in \mathbb{R^+}$. Then $$\frac{d}{dz}g_n\left(z\right)=n\left(1+\frac{z}{n}\right)^{n-1}\cdot\frac{1}{n}=\left(1+\frac{z}{n}\right)^{n-1}$$ Define $g_{\infty}\left(z\right)=\lim _{n\rightarrow \infty }g_n\left(x\right)=\lim_{n\rightarrow \infty }\left(1+\frac{z}{n}\right)^n$, then notice that $$\frac{d}{dz}g_{\infty}\left(z\right)=\frac{d}{dz}\left(\lim _{n\rightarrow \infty }g_n\left(x\right)\right)=\lim _{n\rightarrow \infty }\frac{d}{dz}g_n\left(z\right)$$$$=\lim_{n\to \infty}\left(1+\frac{z}{n}\right)^{n-1}=\lim_{n\to \infty}\frac{\left(1+\frac{z}{n}\right)^n}{1+\frac{z}{n}} =\lim _{n\rightarrow \infty }\left(1+\frac{z}{n}\right)^n=g_\infty\left(z\right)$$ By considering that $g_n\left(0\right)=1\: \forall\:n\in \mathbb{R^+}$, we have the differential equation $$\frac{d}{dz}g_{\infty}\left(z\right)=g_\infty\left(z\right),\,g_\infty \left(0\right)=1$$ Which also has $e^z$ as a solution. However, the above […]

How to solve this limit related to series?

How to solve the following limit? $$\lim_{N\rightarrow+\infty}\frac{1}{N}\sum_{k=1}^{N-1}\left(\frac{k}{N}\right)^N$$

Calculate $\lim_{n\rightarrow \infty} \frac{1}{n}\sum_{k=1}^{n}k\sin\left(\frac{a}{k}\right)$

I’m trying to calculate $\lim_{n\rightarrow \infty} \frac{1}{n}\sum_{k=1}^{n}k\sin\left(\frac{a}{k}\right)$. Intuitively the answer is $a$, but I can’t see any way to show this. Can anyone help? Thanks!

Distance to Cross a City Diagonally

If I had to cross from the southwest corner of a city to the northeast corner of a rectangular city and I could do so by helicopter, the distance would be $\sqrt{x^2 + y^2}$, which is less than $x + y$. If I chose to cross that same city by foot, and I chose to […]

Limit of $a_{k+1}=\dfrac{a_k+b_k}{2}$, $b_{k+1}=\sqrt{a_kb_k}$?

Let $a,b>0$ and let $a_0=a$, $b_0=b $, $a_{k+1}=\dfrac{a_k+b_k} 2$,$b_{k+1}=\sqrt{a_kb_k}$ $\quad k\geq0$. This converges to a number between a and b. Also $a_k>b_k$ for $k\geq1$ (AM-GM inequality). Can we find the limit explicitly in terms of $a$ and $b$?

Confusion with the definition of limit

The definition of limit says that Let $f(x)$ be a function defined on some open interval that contains the number $a$, except possibly at $a$ itself. Then we say that the limit of $f(x)$ as $x$ approaches $a$ is $L$ If….{the rest of definition is left to make the question easier}. What does the phrase […]

Limit of a summation, using integrals method

I have seen an interesting question on stackexchange, which I would like to requote so that I can understand the answer =) $\lim\limits_{n\to\infty} \dfrac{1^{99} + 2^{99} + \cdots + n^{99}}{n^{100}}$ How can the numerator be expressed as an integral? $= \lim\limits_{n\to\infty} \dfrac{1^{99} + 2^{99} + \cdots + n^{99}}{n^{100}}$ $=\lim\limits_{n\to\infty} \frac{\sum\limits_{k=1\to n} k^{99} }{n^{100}}$ $=\lim\limits_{n\to\infty} \frac{\sum\limits_{k=1\to […]

Please prove: $ \lim_{n\to \infty}\sqrt{\frac{1}{n!}} = 0 $

Possible Duplicate: $ \lim\limits_{n \to{+}\infty}{\sqrt[n]{n!}}$ is infinite Please prove: $$ \lim_{n\to \infty}\sqrt[n]{\frac{1}{n!}} = 0 $$