Articles of limits

Existence of a sequence with prescribed limit and satisfying a certain inequality II

This is a more specific variation of the question in the post Existence of a sequence with prescribed limit and satisfying a certain inequality Suppose you have two infinite sequences $\{a_n\},\{b_n\}$, with $0<a_n<b_n < 1$ for each $n$, such that both $a_n, b_n \to 1$ as $n \to \infty$. Further assume that $a_n/b_n \to 1$ […]

Euler-Mascheroni constant: understanding why $\lim_{m\rightarrow \infty} \sum_{n=1}^{m} (\ln (1 + \frac{1}{n})-\frac{1}{n+1})= 1 – \gamma$

I am trying to understand why the Euler-Mascheroni constant $\displaystyle \gamma = \lim_{n \rightarrow \infty} \left ( \sum_{k=1}^n \frac{1}{k} – \ln n \right )$ is equal to $1 – \displaystyle \int_{1}^{\infty} \frac{t-[t]}{t^2}\; \mathrm{d}t$ where $[t]$ is the floor function. There has been an answered question here already: Integral form for the euler-mascheroni gamma constant using […]

limit of $\left( 1-\frac{1}{n}\right)^{n}$

limit of $$\left( 1-\frac{1}{n}\right)^{n}$$ is said to be $\frac{1}{e}$ but how do we actually prove it? I’m trying to use squeeze theorem $$\frac{1}{e}=\lim\limits_{n\to \infty}\left(1-\frac{1}{n+1}\right)^{n}>\lim\limits_{n\to \infty}\left( 1-\frac{1}{n} \right)^{n} > ??$$

Convergence of a recursively defined sequence

Let a sequence $(a_n)_{n=0}^\infty$ be defined recursively $a_{n+1} = (1-a_n)^{\frac1p}$, where $p>1$, $0<a_0<(1-a_0)^{\frac1p}$. Let $a$ be the unique real root of $a=(1-a)^{\frac1p}$, $0<a<1$. It is clear $0<a_0<(1-a_0)^{\frac1p}\Leftrightarrow 0<a_0<a$. Prove 1) $a_{2k-2}<a_{2k}<a<a_{2k+1}<a_{2k-1}$ and $a_{2k+1}-a<a-a_{2k}$. 2) $\lim\limits_{n\to\infty}a_n=a$. Define $f(x):=(1-x)^{\frac1p}$. Consider $f^2$. When $p=2$, $a_{n+2}=f^2(a_n)=\big(1-(1-a_n)^{\frac12}\big)^{\frac12}$. $a_{n+2}>a_n\Leftrightarrow (1-a_n)(1+a_n)^2>1\Leftrightarrow a_n<(1-a_n)^{\frac12}$, and the conclusion is proved. But I am having difficulty […]

How to find $ \sum\limits_{n=0}^{\infty}\frac{1}{3-8n-16n^2}$?

I have to show, that this series converges and determine its limit. $$\sum\limits_{n=0}^{\infty}\frac{1}{3-8n-16n^2}$$ So far, my idea was to transform it into a telescoping series, but I don’t really know how to do it.

Is this claim true for multivariable functions

Today, I am asking to verify the continuity of the following multivariable function: $$f(x,y,z)=\begin{cases}\frac{xz-y^2}{x^2+y^2+z^2}&,\quad(x,y,z)\neq(0,0,0)\\{}\\0&,\quad(x,y,z)=(0,0,0)\end{cases}$$ The continuity can be easily rejected by seeing that the function has no limit at origin when we consider it on two paths: $$(x,0,0), ~~f\to 0 \quad \text{and} \quad (0,y,0), ~~f\to -1$$ As it is clear, this function is homogenous of […]

Limit $\lim_{x\to\infty} x-x^{2}\ln\left(1+\frac{1}{x}\right)$

This question already has an answer here: How do I determine $\lim_{x\to\infty} \left[x – x^{2} \log\left(1 + 1/x\right)\right]$? 4 answers

Is showing $\lim_{z \to \infty} (1+\frac{1}{z})^z$ exists the same as $\lim_{n \to \infty} (1+1/n)^n$ exists

My expanded question: Is showing $\lim_{z \to \infty} (1+\frac{1}{z})^z$ exists as $z$ goes through real values the same as $\lim_{n \to \infty} (1+\frac{1}{n})^n$ exists as $n$ goes through integer values? If not, how much additional work is needed to make the two equivalent? I am asking this because I had posted a question which stated […]

How is $\displaystyle\lim_{n\to\infty}{(1+1/n)^n} = e$?

This question already has an answer here: Evaluating $\lim_{n \to \infty} (1 + 1/n)^{n}$ [duplicate] 4 answers

Question on using sandwich rule with trig and abs function to show that a limit exists.

Past paper Question: For the following function, determine whether $\lim_{x\to\infty}f(x)$ exists, and compute the limit if it exists. Justify your answers. $$f(x)= \dfrac{\sin(x)+1}{\left| x \right|}$$ Attempt: Consider the fact that $-1 \le \sin(x) \le 1$ (for all $x$), which implies $0 \le \sin(x) +1\le 2$. Dividing by $\left| x \right|,$ $$\color{green}{ \frac{0}{\left| x \right|}} \le […]