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 […]

Produce a sequence $(g_n):g_n(x)\ge 0,\,\forall x\in [0,1],\,\forall n\in\Bbb N$ and $\lim g_n(x)\neq 0,\,\forall x\in [0,1]$ but $\int_{0}^{1} g_n\to 0$ Im in need to clarify that Im talking of the Riemann integral. I want some hint or example, Im unable to find a sequence like this. My work at this moment: If the integral converges to […]

Compute $\displaystyle\int_0^\infty \frac{dx}{1+x^3}$ by integrating $\dfrac{1}{1+z^3}$ over the contour $\gamma$ (defined below) and letting $R\rightarrow \infty$. The contour is $\gamma=\gamma_1+\gamma_2+\gamma_3$ where $\gamma_1(t)=t$ for $0\leq t \leq R$, $\gamma_2(t)=Re^{i\frac{2\pi}{3}t}$ for $0\leq t \leq 1$, and $\gamma_3(t)=(1-t)Re^{i\frac{2\pi}{3}}$ for $0\leq t \leq 1$. So, the contour is a wedge, and by letting $R\rightarrow \infty$ we’re integrating over one […]

How to compute this integral? $$ \int^{\pi}_{0} \frac{\sin(nx)\cos\left ( \frac{x}{2} \right )}{\sin \left ( \frac{x}{2} \right ) } \, dx$$ I need your help.

Let $\alpha$ be an irrational number. Then the sequence $\{\{n\alpha\}\}$ is equidistributed. I am using the following definition of equidistribution. A sequence $\{a_i\}$ is equidistributed if $\frac1n\sum\limits_{i=1}^n f(a_i) \to \lambda(f):=\int_0^1 f(x)\,dx$ for all continuous $f:[0,1]\to \mathbb{R}_{\ge 0}$. In the proof I used the fact that if $T_g(x)=(g+x) \pmod 1$, then $\lambda(f\circ T_g)=\lambda(f)$ for all continuous […]

In this answer it states that: $\int_0^1(1-x^4)^{2016}dx=\prod_{j=1}^{2016}\left(1-\frac1{4j}\right)$ How to prove that?

As the title suggests, I want to know the motivation behind the Euler substitutions.How someone arrived at these substitutions. To remove any doubts, by euler substitutions i mean – If we have a rational function of $x$ and $\sqrt{ax^2+bx+c}$ , Case I. If $a>0$ , put $\sqrt{ax^2+bx+c}=t \pm x\sqrt{a}$ Case II. If $c>0$ , put […]

I got this question on my Maths exam today, and the Department of Education has stated it is on the syllabus, but none of the three textbooks I could get my hands mention anything about it. One mentions the volume of a cone, but none the area of a circle. Use integration methods to establish […]

If X~(0,1), integrate to find the moment generating function of a random variable $X^2$ and identify the distribution of $X^2$ using the moment generating function. E[$e^{tx^2}]=\int_{-\infty}^{\infty}{e^{tx^2}e^{-x^2}\over{\sqrt{2\pi}}}$ which reduces to =$1\over{\sqrt{2\pi}}$ $\int_{-\infty}^{\infty}{e^{tx^2}e^{-x^2}}$ =$1\over{\sqrt{2\pi}}$ $\int_{-\infty}^{\infty}{e^{x^2(t-\frac12)}}$ and thus I am stuck. I’m sure there must be some trick to this (like completely the square for the mgf of […]

I would like to compare the change of variable theorem for 1 variable and more. What are the differences, in which case we need stronger assumptions? How do they differ? What is the best way to write the theorems for comparison? Multivariable: Let $\varphi : \Omega \rightarrow \mathbb{R}^n$ (where $\Omega \subset \mathbb{R}^k$ and $k\leq n)$ […]

Intereting Posts

Proving Irrationality
How to prove boundary of a subset is closed in $X$?
Problem about jointly continuous and linearity of expectation.
$AB=BA$ with same eigenvector matrix
Distributive Law and how it works
Is it true that all single variable integral that had closed form can solve by one algorithms?
Computation of $\int _{-\pi} ^\pi \frac {e^{in\theta} – e^{i(n-1)\theta}} {\mid \sin {\theta} \mid} d\theta .$
show that a distance function is continuous
$\int f_n g \, d\mu \to \int fg \, d\mu$ for all $g$ which belongs to $\mathscr{L}^q (X)$ (exercise)
How can I prove this relation between the elementary set theory and the elementary logic?
Axiomatic approach to polynomials?
Overview of basic results about images and preimages
Is $AA^T$ a positive-definite symmetric matrix?
Prove that there is no element of order $8$ in $SL(2,3)$
Meaning of “a mapping preserves structures/properties”