Tagged Questions
2
votes
1answer
62 views
Solving the equation $\displaystyle \frac{e^x}{x}=\int_n^{n+1}f(t)\,dt$
Suppose the equation $\displaystyle \frac{e^x}{x}=\int_n^{n+1}f(t)\,dt$ as $f(t)=\frac{e^t}{t}$ and $n\in \mathbb{N} \setminus{0}$.
How to prove that:
The equation above has a unique solution $U_n$ ...
4
votes
3answers
99 views
How to prove that $\lim\limits_{n\to\infty}\int\limits _{a}^{b}\sin\left(nt\right)f\left(t\right)dt=0\text { ? }$
Let $f:\left[a,b\right]\to\mathbb{R}$ be a function that is derivative so that $f'$ is continuous then
$$
\lim_{n\to\infty}\int\limits _{a}^{b}\sin\left(nt\right)f\left(t\right)dt=0
$$
My attempt: I ...
1
vote
2answers
29 views
Existence of limit of $\lim_{n \to \infty}\sum_{i=0}^{[n/2]} \frac 1 n f \left(\frac i n \right)$!
If $f$ is continuous in $[0,1]$ then $$\lim_{n \to \infty}\sum_{i=0}^{[n/2]} \frac 1 n f \left(\frac i n \right)$$ (where $[y]$ is the largest integer less than or equal to $y$)
(A) does not ...
2
votes
5answers
103 views
Evaluate $ \lim\limits_{x \to 0} \frac{1}{\sin^3x}\int_0^x{\sin(t^2) } dt$
$$ \lim_{x \to 0} \frac{1}{\sin^3x}\int_0^x{\sin(t^2) dt}$$
This is what I've tried:
Let $F(x) = \displaystyle\int_0^x{\sin(t^2) dt}$, and let $f(x) = {\sin(t^2)}$.
Then $F'(x) = f(x) ...
1
vote
1answer
47 views
The integral $\int_0^1\dfrac{(-x)^n}{1+x} dx $
How can I prove that:
$\forall x \in \mathbb{N}\setminus {0} \quad \dfrac{-1}{n+1}\le \int_0^1\dfrac{(-x)^n}{1+x} dx \le \dfrac{1}{1+n}$
$\lim_{n\to+\infty}\Sigma_{i=1}^{n}\dfrac{(-1)^{i-1}}{i}$.
...
1
vote
1answer
91 views
Can the integral of $x^x$ be found?
I'm interested in knowing if the indefinite integral of $x^x$ can be found in terms of elementary functions.
I am under the impression (be it correct or incorrect) that it can be found. This is why: ...
0
votes
1answer
52 views
am I doing this integration by parts right
I am supposed to show that $-\int_{-\infty}^a f(t)dt =\int_{-\infty}^a (t-a)f'(t)dt$ where $f$ is the cdf of a RV.
Now what I do is $RHS= \int_{-\infty}^a tf'(t)dt-a\int_{-\infty}^a ...
2
votes
1answer
43 views
lim of integration of a non-negative function.
i'd like very much your help with this one :
given a positive function meaning
$$ f(x) \geq 0 $$
and $f$ is continuous.
Let $$ M = \sup(f(x)) $$ where $x$ belongs to $[a,b]$.
How can i prove that ...
2
votes
2answers
58 views
Two integration problems
Find out if this limit exists: $$\lim_{y\to 0}\int_0^1\frac{x e^{\frac{-x^2}{y^2}}}{y^2}dx$$
Evaluate $$\int_0^{\pi/2}\dfrac{\arctan(a \tan(x))}{\tan(x)}$$
I'm a bit lost with these two problems. I ...
4
votes
2answers
80 views
Let $f>0$ differentiable in $[0,\infty)$. Assume $\lim \limits_{x \to \infty} (\log\circ f)^\prime(x) < 0$. Show that $\int_0^\infty f$ converges.
So what I gathered from the givens about $f$, since $(\log\circ f)^\prime(x)=\frac{f^\prime(x)}{f(x)}$ it would mean that far enough, $f^\prime(x)<0$. I don't know how to go about this from here.
...
3
votes
1answer
62 views
When does $\lim\limits_{n\to\infty}\int_{b}^{a_n}f_n(x)dx=\lim\limits_{n\to\infty}\int_b^\infty f_n(x)dx$ hold?
Let $\{a_n\}\subset \mathbb{R}$ be sequence and $$f_n:[b,\infty)\longrightarrow \mathbb{R}, \qquad n=1,2,\dots .$$
Assume that $$\lim_{n\longrightarrow\infty}a_n=+\infty.$$
Obviously, from the ...
1
vote
1answer
19 views
Symmetric limits
I read that you can use something called a "symmetric limit" to evaluate the improper integral of sin(x) as 0, something that I attempted by taking the right hand improper integral and adding it to ...
1
vote
1answer
41 views
Lebesgue integrable function and limit
Show that if $f$ is a Lebesgue integrable function on $A\subset\mathbb R$ and $$A_n=\{x\in A:|f(x)|\geq n\}$$ for $n\in\mathbb N$, then $\lim_{n\to\infty} n\cdot m(A_n)=0$.
My solution which is ...
9
votes
3answers
136 views
Evaluating $\lim \limits_{n\to \infty} \left( n \int_{0}^{\frac \pi 2} 1-\sqrt [n]{\sin x} \,\mathrm dx \right)$
Evaluate the following limit:
$$\lim \limits_{n\to \infty} \;\; n \int_{0}^{\frac \pi 2} \left(1-\sqrt [n]{\sin x} \right)\,\mathrm dx $$
I have done the problem .
How I solved is
First I ...
5
votes
1answer
70 views
Prove that $\lim_{t \rightarrow 0} t \int_{0}^{\infty} e^{-tx} f(x) dx =1$
I am trying to solve Rudin 8.11:
Suppose $f$ is Riemann-integrable on $[0,A]$ for all $A<\infty$, and $f(x) \rightarrow 1$ as $x \rightarrow \infty$.
Prove that
$$\lim_{t \rightarrow 0} ...
2
votes
0answers
36 views
Is this pointwise convergence sequence also uniform convergence?
$f_{n}$ and $f$ are continuous functions and $f_{n}\rightarrow f$ pointwise. Which of the following are correct?
$\int _{0}^{x}F_{n}\left( t\right) dt\rightarrow\int _{0}^{x}F\left( t\right) dt$
...
1
vote
1answer
112 views
$f_n$ converges pointwise to $f$ implies integral $f_n$ converges to integral $f$
Let $\lambda(E)< \infty$ and $f_n$ be measurable and continuous (on $E$) for each $n\in\mathbb{N}$.
If $f_n$ converges pointwise to $f$ (continuous on $E$) for all $x\in E$, then $\int_Ef_n ...
1
vote
1answer
42 views
Improper integrals determine if they converge or diverge.
The question is as follows.
Determine if the these 2 improper integrals converge.
$\int^{\infty}_{0} ( x^{1/2} +x^{3/2} )^{-1}$ And $\int^{\pi}_{0} (1-\cos(x))/(\sin^{2}(x))$
For the first ...
1
vote
2answers
99 views
Computing $\lim_{n \rightarrow\infty} \int_{a}^{b}\left ( f(x)\left | \sin(nx) \right | \right )$ with $f$ continuous on $[a,b]$
Let $a,b \in \mathbb{R}$ and $\textit{f} :[a,b] \rightarrow \mathbb{R}$ continuous on $[a,b]$.
Does the sequence $\left (\int_{a}^{b} f(x)\left |\sin(nx) \right |dx \right )$ converge? If it does, ...
1
vote
2answers
62 views
Calculus, integration, Riemann sum help?
Express as a definite integral and then evaluate the limit of the Riemann sum lim
$$
\lim_{n\to \infty}\sum_{i=0}^{n-1} (3x_i^2 + 1)\Delta x,
$$
where $P$ is the partition with
$$
x_i = -1 + ...
3
votes
1answer
59 views
Prove the existence of a limit : $ \lim_{x\rightarrow+\infty}{\int_{\varepsilon}^{+\infty}{xF(xt)\cos{t}dt}}=0$
Let $F(x),G(x)$ be nonnegative decreasing functions in $[0,+\infty)$, with$\,\displaystyle \lim_{x\rightarrow+\infty}{x(F(x)+G(x))}=0$
(1) Prove that: $\forall \varepsilon>0$,we have ...
2
votes
1answer
127 views
A multiple integral question II
We know from the previous post that
$$\lim_{n\to\infty}\underbrace{\int_0^1 \int_0^1 \cdots \int_0^1}_{n \text{ times}}\frac{1}{(x_1\cdot x_2\cdots x_n)^2+1} ...
6
votes
1answer
91 views
A multiple integral question
Proving that
$$\lim_{n\to\infty}\underbrace{\int_0^1 \int_0^1 \cdots \int_0^1}_{n \text{ times}}\frac{1}{(x_1\cdot x_2\cdots x_n)^2+1} \mathrm{d}x_1\cdot\mathrm{d}x_2\cdots\mathrm{d}x_n=1$$
2
votes
2answers
177 views
Prove an integral limit
Let $F(x),G(x)\ge 0$ be decreasing functions on $[0,+\infty)$
and
$\displaystyle\lim_{x\to+\infty}x(F(x)+G(x))=0$
(1) Prove that: ...
3
votes
1answer
98 views
Integral with Undefined Endpoint (Complex Variables)
The problem is find: $\int\limits_0^1 \lim\limits_{n\rightarrow\infty}(nz^{n-1})dz$
I started by finding $\lim\limits_{n\rightarrow\infty}(nz^{n-1})$. Naturally it converges to zero on [0,1). However ...
0
votes
1answer
167 views
Finding limits of integration in convolution
I am struggling to fully get how to choose proper limits of integration when calculating convolutions. Right now I am stuck on a problem where I have to show that when taking the Fourier transform of ...
0
votes
1answer
93 views
Application of Dominated Convergence Theorem.
Find $L=\lim\limits_{n \to \infty} \int_0^{n a} \exp\left(-\dfrac{t}{1+\frac{b t}{n}}\right) dt$, where $a>0$, $b>0$.
I can't see what is dominating function, but I feel that I have to use ...
5
votes
2answers
55 views
If $f_n \in L^1$, will the limit function $f$ also be in $L^1$ in monotone convergence theorem?
Monotone convergence theorem doesn't require the sequence of functions $f_n$'s to be $L^1$. When $f_n\in L^1$, will its pointwise limit function $f$ also be in $L^1$? Thanks!
0
votes
1answer
182 views
Comparison Theorem for Integral Calculus
I have narrowed it down to C, E, and F, since we know that $1/x^{1/5}$ is always greater than the original function for all $x\geq 1$. However, the second set of conditions is more difficult to ...
3
votes
1answer
79 views
A question about a limit which involve Riemman integeral
Let $f\in C^{1}[0,1]$,is this true?
\begin{align*}
\lim_{n\to\infty}\frac{\int_{\frac{i}{n}}^{\frac{i+1}{n}}f(x)dx-\frac{1}{2n}[f(\frac{i}{n})+f(\frac{i+1}{n})]}{\frac{1}{n^2}}=0
\end{align*}
...
0
votes
1answer
92 views
if $f(x)=o(g(x))$ will $\int_0^x f(x)=o(\int_0^x g(x))$ and $f'(x)=o(g'(x))$?
Let $f$ and $g$ be two functions with derivatives in some interval containing $0$, where $g$ is positive. Also
$$f(x)=o(g(x))~as~x \rightarrow0$$
Prove or dissprove:
1) ...
2
votes
2answers
67 views
Evaluate: $\lim_{h \to 0} \int_{-1}^{1}\frac{h}{h^2+x^2}~dx$
How can I evaluate: $$\lim_{h \to 0} \int_{-1}^{1}\frac{h}{h^2+x^2}~dx$$
How I proceed:
$$\lim_{h \to 0} \int_{-1}^{1}\frac{h}{h^2+x^2}~dx=2\lim_{h \to 0} ...
4
votes
1answer
134 views
$f$ continous at $x_0$ $⇒\lim_{h→0}∫_{x_0}^{x_0+h}\frac{f(t)}{h}=f(x_0)$
The function $f:ℝ→ℝ$ is continuous on $x_0\inℝ$.
Prove using the definition of a Darboux Integral that
$$\lim_{h→0}∫_{x_0}^{x_0+h}\frac{f(t)}{h}=f(x_0)$$
I'm a first grade math student following an ...
1
vote
0answers
91 views
$\lim_{n\to\infty}\int_{0}^{2\pi}\sin(nx)f(x)dx=?$ [duplicate]
Possible Duplicate:
How can I prove $\lim_{n \to \infty} \int_{0}^{\pi/2} f(x) \sin ((2n+1) x) dx =0 $?
I have trouble evaluating the following limit?
$$
...
5
votes
3answers
220 views
Limit of $x\log x\cdot f(x)$ when $f$ is integrable function
Suppose that $f\in L^1(0,+\infty)$ is a monotone decreasing, positive function. Prove that $\lim_{x \to +\infty}x(\log x)\cdot f(x)=0.$
1
vote
3answers
118 views
Question about liminf for a pointwise convergent sequence of functions.
If $f_n \rightarrow f$ pointwise, then does $$\liminf \int f_n=\lim\int f_n?$$
I know that $\liminf f_n=\lim f_n$ since the sequence converges, but I'm not sure if the $(L)$ integral throws us off.
...
2
votes
1answer
56 views
Integration limits when integrating both sides
I have been working on solving differential equations and this is really cracking me up.
I obtained the following equation:
dz/dr = r
and I wish to obtain z in terms of r, given that we know that z ...
0
votes
3answers
69 views
Does a function sequence with the integral 1 have a limit with the integral 1?
Let $(x_n)$ be any function sequence such that
$$ \int_0^1x_n(t)dt=1 \qquad \forall n $$
$$ \lim_{n\to\infty}x_n = x $$
I'm trying to prove that the limit $x$ also has the property ...
3
votes
3answers
136 views
Evaluate $\lim_{n \to \infty}\int^n_1 \frac{\left |\sin x \right |}{n}dx$
Evaluate $$\lim_{n \to \infty}\int^n_1 \frac{|\sin x|}{n}dx$$
I think that I should deal with $\int|\sin x|dx$, but I don't know how to go on. Please help. Thank you.
1
vote
4answers
263 views
Integration with infinity and exponential
How is
$$\lim_{T\to\infty}\frac{1}T\int_{-T/2}^{T/2}e^{-2at}dt=\infty\;?$$
however my answer comes zero because putting limit in the expression, we get:
$$\frac1\infty\left(-\frac1{2a}\right) ...
0
votes
1answer
236 views
How to solve Fourier Series
How can I easily solve fourier series equations given below? I want to solve it quickly and easily in exams cuz I am not good with mathmatics very much..
For Example:
$a_n= \int_0^\pi e^{-t/2} ...
0
votes
3answers
175 views
Proving the limit of an improper integral of a sequence of functions.
I was trying to prove that the following limit
$$\lim_{n\to\infty}\int^{\infty}_{1}\frac{\sin{x}}{x^{n+1}}\mathrm{d}x$$
is equal to $0$. I believe that the easiest option in similar cases - and the ...
10
votes
4answers
261 views
Is this limit equal to 1?
Let $x\in R^{n}$ be fixed ($x\neq 0$) and $r>0$. Evaluate the limit:$$\lim_{r\rightarrow \infty}\frac{V(B(x,r)\cap B(0,r))}{V(B(0,r))}$$
where $V$ stands for volume and $B(x,r)$ is the ball with ...
0
votes
3answers
318 views
Solving improper integrals and u-substitution on infinite series convergent tests
This is the question:
Use the integral test to determine the convergence of $\sum_{n=1}^{\infty}\frac{1}{1+2n}$.
I started by writing:
$$\int_1^\infty\frac{1}{1+2x}dx=\lim_{a \rightarrow ...
4
votes
3answers
108 views
Limit of Integral of square of function
Let $f: [0,1)\rightarrow \mathbb{R}$ be continuous and nonnegative on $[0,1)$. Prove that if $\lim_{a\rightarrow 1^{-}}{\int_{0}^{a}{f(x)^{2}dx}}$ exists, then $\lim_{a\rightarrow ...
2
votes
1answer
139 views
Order of integration
I am reading a book by L. D. Landau titled Mechanics and there is a "changing order of the integral" step on page 28 that I don't get:
$$\int_0^a\int_0^E \left[{dx_2\over dU}-{dx_1\over ...
7
votes
1answer
162 views
Calculate: $\lim_{n\to\infty} \int_{0}^{\pi/2}\frac{1}{1+x\tan^{n} x }dx$
I'm supposed to work out the following limit:
$$\lim_{n\to\infty} \int_{0}^{\pi/2}\frac{1}{1+x \left( \tan x \right)^{n} }dx$$
I'm searching for some resonable solutions. Any hint, suggestion is ...
7
votes
2answers
198 views
Evaluating: $\lim_{n\to\infty} \frac{\sqrt n}{\sqrt {2}^{n}}\int_{0}^{\frac{\pi}{2}} (\sin x+\cos x)^n dx $
I'm supposed to compute the following limit:
$$\lim_{n\to\infty} \frac{\sqrt n}{\sqrt {2}^{n}}\int_{0}^{\frac{\pi}{2}} (\sin x+\cos x)^n dx $$
I'm looking for a resonable approach in this case, if ...
5
votes
2answers
160 views
Alternative proof of the limitof the quotient of two sums.
I found the following problem by Apostol: Let $a \in \Bbb R$ and $s_n(a)=\sum\limits_{k=1}^n k^a$. Find
$$\lim_{n\to +\infty} \frac{s_n(a+1)}{ns_n(a)}$$
After some struggling and helpless ideas I ...
8
votes
3answers
643 views
Computing $\lim\limits_{n\to\infty} \int_{0}^{\infty}\frac{\sin(x/n)}{(1+x/n)^{n}}\, dx$
$$\lim_{n\to\infty} \int_{0}^{\infty}\frac{\sin(x/n)}{(1+x/n)^{n}}\, dx$$
I've been able to show that the integral is bounded above by 1 (several ways).
One of the simplest is just letting $u=x/n$ ...
