Articles of limits

How is this series for the Euler-Mascheroni constant derived?.

We are familiar with the classic sum for Euler’s constant $\gamma$: $$ \gamma=\lim_{n\to \infty}\left(\sum_{k=1}^{n}\frac{1}{k}-\ln(n)\right) .$$ But, how is this one derived?: $$ \gamma=\lim_{n\to \infty}\left(\frac12\sum_{k=1}^{\infty}\frac{(-1)^{k+1}}{k(2k)!}n^{2k}-\ln(n)\right) .$$ I thought perhaps it may have something to do with the Bernoulli numbers or even Zeta because there are terms which appear in their identities (such as the formula for […]

Find $\lim_{x\to0}\frac{\sin5x}{\sin4x}$ using $\lim_{\theta\to0}\frac{\sin\theta}{\theta}=1$.

I am trying to find $$\lim_{x\to0}\frac{\sin5x}{\sin4x}$$ My approach is to break up the numerator into $4x+x$. So, $$\begin{equation*} \lim_{x\to0}\frac{\sin(4x+x)}{\sin4x}=\lim_{x\to0}\frac{\sin4x\cos x+\cos4x\sin x}{\sin4x}\\ =\lim_{x\to0}(\cos x +\cos4x\cdot\frac{\sin x}{\sin4x})\end{equation*}$$ Now the problem is with $\frac{\sin x}{\sin4x}$. If I use the double angle formula twice, it is going to complicate the problem. The hint says that you can use $\lim_{\theta\to0}\frac{\sin\theta}{\theta}=1$. […]

Finding $\lim\limits_{x \to 0}\ \frac{\sin(\cos(x))}{\sec(x)}$

The problem is to find: $\lim\limits_{x \to 0}\ \dfrac{\sin(\cos(x))}{\sec(x)}$ I rewrite the equation as follows: $\lim\limits_{x \to 0}\ \dfrac{\sin(\cos(x))}{\dfrac{1}{\cos(x)}}$ And multiply by $\dfrac{\cos(x)}{\cos(x)}$, producing: $\lim\limits_{x \to 0}\ \dfrac{\cos(x)*\sin(\cos(x))}{\dfrac{\cos(x)}{\cos(x)}}$ And rewrite as: $\lim\limits_{x \to 0}\ \cos^2(x)\ \dfrac{\sin(\cos(x))}{\cos(x)}$ Which then becomes: $\lim\limits_{x \to 0}\ \cos^2(x) * 1$ Which becomes 1. However, the answer is apparently $\sin(1)$. What […]

Limit of an integral (remainder term of a Euler-Maclaurin expansion)

For every $x > 0$, define $$I(x) = \int_1^\infty \left(\{t\} – \frac{1}{2}\right)\frac{x}{e^{xt}-1}\,dt.$$ where $\{x\} = x – \lfloor x\rfloor$ denotes the fractional part of $x$. How to justify that $I(x)$ converges to the improper integral $$ \int_1^\infty \left(\{t\}-\frac{1}{2}\right)\frac{dt}{t} $$ when $x \searrow 0$ ? Of course $\dfrac{x}{e^{tx}-1}$ converges to $\dfrac{1}{t}$ for every $t$, but since […]

Estimate of $n$th prime

There is a result that if $p_n$ is the $n$th prime, then $p_n\sim n\log n$ as $n\rightarrow\infty$. I wonder: Is it a direct consequence of the prime number theorem $\pi(x)\sim x/\log x$? The theorem says that there are approximately $n/\log n$ primes less than or equal to $n$. So there are approximately $n$ primes less […]

Limit $\lim_{x\to 0}\left(\frac{1}{1^\left (\sin^2x \right)}+\cdots+\frac{1}{n^\left (\sin^2x \right)}\right)^\left(\sin^2x \right) = ?$

$$\lim_{x\to 0}\left(\frac{1}{1^\left (\sin^2x \right)}+\frac{1}{2^\left (\sin^2x \right)}+\cdots+\frac{1}{n^\left (\sin^2x \right)}\right)^\left(\sin^2x \right) = ?$$ This doesn’t seem an indeterminate form to me so I directly substituted the value of x to evaluate the limit. Am I right?

Calculating $\lim\limits_{C \rightarrow \infty} -\frac{1}{C} \log(1 – p \sum\limits_{k=0}^{C}e^{-\gamma C} (\gamma C)^k/k!) $

How to calculate the following limit: $$\lim_{C\rightarrow \infty} -\frac{1}{C} \log\left(1 – p \sum_{k=0}^{C}\frac{e^{-\gamma C} (\gamma C)^k}{k!}\right)$$ Given that $0 \leq \gamma \leq 1$ and $0 \leq p \leq 1$. At least any tips about approaching the solution!

Does the little-oh relation remain if $f(x)$ and $g(x)$ both integrate or differentiate?

Give two functions $f$ and $g$ with derivatives in some interval containing 0,where $g$ is positive.Assume also $f(x)=o(g(x))$ as $x \to 0$. Prove or disprove each of the following statements: (a) $\int^x_0 f(t)dt = o(\int^x_0g(t)dt)$ as $x \to 0$ (b) $f'(x)=o(g'(x))$ as $x \to 0$ I use the definition to prove (a) $$\begin{align} \lim_{x \to […]

Find limit $a_{n+1} = \int\limits^{a_n}_0 \bigl( 1+\frac 1 4 \cos^{2n+1}t \bigr) \, dt$

Let $a_0 \in (0, 2\pi)$. Find limit $$a_{n+1} = \int\limits^{a_n}_0 \bigl( 1+\frac 1 4 \cos^{2n+1}t \bigr) \, dt$$

Limit of Multivariate Probability Density Function as one or more or all variables approach positive or negative infinity

Could someone point out a good source or provide justification as to whether the limit of a multivariate probability density function, as one or more or all variables tends to positive or negative infinity, goes to zero. $$\lim_{x_{1}\to\pm\infty}f(x_{1},x_{2},…,x_{n})=0$$ $$\lim_{x_{1},x_{2}\to\pm\infty}f(x_{1},x_{2},…,x_{n})=0$$ $$\lim_{x_{1},x_{2},…,x_{n}\to\pm\infty}f(x_{1},x_{2},…,x_{n})=0$$ I have found this related question for the univariate case: Limit of probability density function […]