Show that $$ \lim_{n \to \infty} \int\limits_{0}^{h} \frac{\sin (n\varepsilon)}{\varepsilon} \;\mathrm{d}\varepsilon = \int\limits_{0}^{\infty} \frac{\sin (t)}{t} \;\mathrm{d}t, \;\;h>0 $$
2026-04-13 04:41:46.1776055306
limits and integrals
56 Views Asked by Bumbble Comm https://math.techqa.club/user/bumbble-comm/detail At
1
There are 1 best solutions below
Related Questions in INTEGRATION
- How can I prove that $\int_0^{\frac{\pi}{2}}\frac{\ln(1+\cos(\alpha)\cos(x))}{\cos(x)}dx=\frac{1}{2}\left(\frac{\pi^2}{4}-\alpha^2\right)$?
- How to integrate $\int_{0}^{t}{\frac{\cos u}{\cosh^2 u}du}$?
- Show that $x\longmapsto \int_{\mathbb R^n}\frac{f(y)}{|x-y|^{n-\alpha }}dy$ is integrable.
- How to find the unit tangent vector of a curve in R^3
- multiplying the integrands in an inequality of integrals with same limits
- Closed form of integration
- Proving smoothness for a sequence of functions.
- Random variables in integrals, how to analyze?
- derive the expectation of exponential function $e^{-\left\Vert \mathbf{x} - V\mathbf{x}+\mathbf{a}\right\Vert^2}$ or its upper bound
- Which type of Riemann Sum is the most accurate?
Related Questions in LIMITS
- How to prove $\lim_{n \rightarrow\infty} e^{-n}\sum_{k=0}^{n}\frac{n^k}{k!} = \frac{1}{2}$?
- limit points at infinity
- Calculating the radius of convergence for $\sum _{n=1}^{\infty}\frac{\left(\sqrt{ n^2+n}-\sqrt{n^2+1}\right)^n}{n^2}z^n$
- Maximal interval of existence of the IVP
- Divergence of power series at the edge
- Compute $\lim_{x\to 1^+} \lim_{n\to\infty}\frac{\ln(n!)}{n^x} $
- why can we expand an expandable function for infinite?
- Infinite surds on a number
- Show that f(x) = 2a + 3b is continuous where a and b are constants
- If $a_{1}>2$and $a_{n+1}=a_{n}^{2}-2$ then Find $\sum_{n=1}^{\infty}$ $\frac{1}{a_{1}a_{2}......a_{n}}$
Related Questions in FOURIER-ANALYSIS
- An estimate in the introduction of the Hilbert transform in Grafakos's Classical Fourier Analysis
- Verifying that translation by $h$ in time is the same as modulating by $-h$ in frequency (Fourier Analysis)
- How is $\int_{-T_0/2}^{+T_0/2} \delta(t) \cos(n\omega_0 t)dt=1$ and $\int_{-T_0/2}^{+T_0/2} \delta(t) \sin(n\omega_0 t)=0$?
- Understanding Book Proof that $[-2 \pi i x f(x)]^{\wedge}(\xi) = {d \over d\xi} \widehat{f}(\xi)$
- Proving the sharper form of the Lebesgue Differentiation Theorem
- Exercise $10$ of Chapter $4$ in Fourier Analysis by Stein & Shakarchi
- Show that a periodic function $f(t)$ with period $T$ can be written as $ f(t) = f_T (t) \star \frac{1}{T} \text{comb}\bigg(\frac{t}{T}\bigg) $
- Taking the Discrete Inverse Fourier Transform of a Continuous Forward Transform
- Is $x(t) = \sin(3t) + \cos\left({2\over3}t\right) + \cos(\pi t)$ periodic?
- Translation of the work of Gauss where the fast Fourier transform algorithm first appeared
Related Questions in FOURIER-SERIES
- order of zero of modular form from it's expansion at infinity
- Fourier series expansion of $\frac{\pi^4}{96}$ and $\frac{\pi^4}{90}$
- How is $\int_{-T_0/2}^{+T_0/2} \delta(t) \cos(n\omega_0 t)dt=1$ and $\int_{-T_0/2}^{+T_0/2} \delta(t) \sin(n\omega_0 t)=0$?
- Fourier series. Find the sum $\sum_{n=1}^\infty \frac{(-1)^{n+1}}{2n+1}$
- How get a good approximation of integrals involving the gamma function, exponentials and the fractional part?
- The convolution theorem for fourier series.:$ \widehat{f*g}(x) =2π\hat{g}(x)\cdot\hat{f}(x) $
- Ergodicity of a skew product
- Fourier Series on $L^1\left(\left[0,1\right)\right)\cap C\left(\left[0,1\right)\right)$
- Parseval's Identity Proof Monotone/Dominated Convergence Theorem
- How can I interchange the sum signs
Trending Questions
- Induction on the number of equations
- How to convince a math teacher of this simple and obvious fact?
- Find $E[XY|Y+Z=1 ]$
- Refuting the Anti-Cantor Cranks
- What are imaginary numbers?
- Determine the adjoint of $\tilde Q(x)$ for $\tilde Q(x)u:=(Qu)(x)$ where $Q:U→L^2(Ω,ℝ^d$ is a Hilbert-Schmidt operator and $U$ is a Hilbert space
- Why does this innovative method of subtraction from a third grader always work?
- How do we know that the number $1$ is not equal to the number $-1$?
- What are the Implications of having VΩ as a model for a theory?
- Defining a Galois Field based on primitive element versus polynomial?
- Can't find the relationship between two columns of numbers. Please Help
- Is computer science a branch of mathematics?
- Is there a bijection of $\mathbb{R}^n$ with itself such that the forward map is connected but the inverse is not?
- Identification of a quadrilateral as a trapezoid, rectangle, or square
- Generator of inertia group in function field extension
Popular # Hahtags
second-order-logic
numerical-methods
puzzle
logic
probability
number-theory
winding-number
real-analysis
integration
calculus
complex-analysis
sequences-and-series
proof-writing
set-theory
functions
homotopy-theory
elementary-number-theory
ordinary-differential-equations
circles
derivatives
game-theory
definite-integrals
elementary-set-theory
limits
multivariable-calculus
geometry
algebraic-number-theory
proof-verification
partial-derivative
algebra-precalculus
Popular Questions
- What is the integral of 1/x?
- How many squares actually ARE in this picture? Is this a trick question with no right answer?
- Is a matrix multiplied with its transpose something special?
- What is the difference between independent and mutually exclusive events?
- Visually stunning math concepts which are easy to explain
- taylor series of $\ln(1+x)$?
- How to tell if a set of vectors spans a space?
- Calculus question taking derivative to find horizontal tangent line
- How to determine if a function is one-to-one?
- Determine if vectors are linearly independent
- What does it mean to have a determinant equal to zero?
- Is this Batman equation for real?
- How to find perpendicular vector to another vector?
- How to find mean and median from histogram
- How many sides does a circle have?
Hint: Let $t = n\epsilon $. ${}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}$