For basic questions about limits, derivatives, integrals, and applications, mainly of one-variable functions.

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12 views

Can someone suggest some reference for a particular kind of infinite series

Can someone suggest some reference for the properties of the series $\lim_{n \to \infty} \sum_{i = 0}^{n} f(n, i)$ For example, when can we apply approximations to $f(n, i)$ that only works when $n ...
0
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0answers
8 views

Will this technique always find the maximum of a positive-definite function?

Definitions: A real-valued, continuously differentiable function $f$ is positive definite on a neighborhood of the origin, $D$, if $f(0) = 0$ and $f(x) > 0$ for every non-zero $x \in D$. ...
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0answers
16 views

How do you calculate an area enclosed by four tangents by using the integration method?

For example, make it $y=3x-6$, $y=3x-15.48$, $y=-0.25x+1.25$, and $y=-0.25x-1.06$. It's been taken by finding the tangent line of a curve $y=(x-2)(x-3)(x-5)$.
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0answers
12 views

A question about fixed-point iteration sequence of a two times continuously differentiable function

I am stuck at this problem: Let $g:[a,b]\to[a,b]$ be a 2 times continuously differentiable function that satisfy: for all $x\in [a,b]$, $g''(x)\neq 0$ And let $s$ be an arbitrary fixed-point of ...
2
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2answers
53 views

how to prove that $ \lim x_n^{y_n}=\lim x_n^{\lim y_n}$?

Let $(x_n)$ and $(y_n)$ be two sequences of real numbers, such that: $\lim x_n = a > 0$ and $\lim y_n = b \in \mathbb R$ I need to prove that: $$\lim \left( x_n ^{y_n} \right) = a^b$$ I tried ...
1
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4answers
51 views

evaluate $\lim_{n\to\infty}\sum_{r=1}^{n-1}\frac{e^{r/n}}{n}$

Each term in the equation given in title tends to zero. $e^{\frac{r}{n}}$ tends to 1 and the denominator tends to infinity. Also, even the greatest numerator in the summation $e^{\frac{n-1}{n}}$ ...
2
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2answers
46 views

GRE Math Subject Practice Test Question $53$

f and g are function of a real variable such that $g(x) = \int_0^x f(y)(y-x)dy$ for all $x$. If g is three times continuously differentiable, what is the greatest integer n for which f must be $n$ ...
1
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0answers
17 views

simplifying complex expression

Hi I am trying to simplify the following expression:$$ \left|\frac{1}{a+ib}\left(\frac{J_1(c x)}{J_1(c b)}-x\right)\right|^2,\quad a,b,x\in \mathbb{R}, \ c\in \mathbb{C} $$ Is there a simple way of ...
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0answers
24 views

Integral $\int x^2\Re(J_1(ax))dx$

$$ \int x^2 \, \Re\left[{J_1(a x)}\right]dx,\quad a\in \mathbb{C}. $$ This integral cannot be done in terms of elementary functions, and since it's $x\cdot J_1(ax)$ we cannot reduce it to other ...
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63 views

Could it possibly have a nice closed form? $\int _0^1\int _0^1\frac{x y}{(x+1) (y+1) \log (x y)}\ dx \ dy$

Using multiple integrals it's not hard to show that the present integral reduces to some integral over squared digamma functions, but then things become harder. How would you tackle the problem? ...
1
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1answer
16 views

Understanding step in derivation of joint distribution

In a derivation I am trying to understand, there is the following argument: \begin{align} &=\int n!\prod_{i=1}^n f_X(x_i)\mathbb{I}_{x_1\le x_2\le\ldots\le ...
1
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1answer
16 views

How to calculate the partition function of a given distribution?

As noted in A FULL BAYESIAN APPROACH FOR INVERSE PROBLEMS, let $ y = Ax + n$, where $x$ is a $m$ dimensional signal and $n$ is white Gaussian noise with precision $\beta$, so we have: $$ y|x, \beta ...
2
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2answers
22 views

Show that any 2D vectors can be expressed in the form…

(a) Show that any 2D vector can be expressed in the form $s \begin{pmatrix} 3 \\ -1 \end{pmatrix} + t \begin{pmatrix} 2 \\ 7 \end{pmatrix},$ where $s$ and $t$ are real numbers. (b) Let $u$ and $v$ be ...
1
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1answer
46 views

Condition on p for convergence of $\sum{\frac{1}{n(\log(n))^p}}$

For what values of $p$ is the series $\sum{\frac{1}{n(\log(n))^p}}$ divergent and for what values it is convergent?
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3answers
70 views

Series convergence $x+\frac{2^2x^2}{2!}+\frac{3^3x^3}{3!}+\frac{4^4x^4}{4!}+\cdots$ [on hold]

Choose the right option. The series $x+\dfrac{2^2x^2}{2!}+\dfrac{3^3x^3}{3!}+\dfrac{4^4x^4}{4!}+\cdots$ is convergent if a. $0<x<1/e$ b. $x>1/e$ c. $2/e<x<3/e$ d. $3/e<x<4/e$ ...
4
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2answers
66 views

Problem 7 IMC 2015 - Integral and Limit

I'm trying to solve problem 7 from the IMC 2015, Blagoevgrad, Bulgaria (Day 2, July 30). Here is the problem Compute $$\large\lim_{A\to\infty}\frac{1}{A}\int_1^A A^\frac{1}{x}\,\mathrm dx$$ ...
-5
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2answers
36 views

solve this linear equation

Using linear differential equation, solve the following equation $( y \log (x)-2) y \textrm{d} x =x \textrm{d}y$. Source: "higher engineering mathematics by grewal"
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0answers
25 views

Proof that the sum of a certain infinite series can be bounded to zero

$\forall 0 < \alpha < 1$, there exists $\lambda > 0$, $k > 0$, s.t. $$ \lim_{n \to \infty} \sum_{w = 1}^{\lambda n} \binom{n}{w} \frac{1}{2^{\alpha n}}\left(1 +\left(1 - ...
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3answers
66 views

Does l'Hopital rule work for -inf/inf?

If you have an indeterminate form: $\frac{-\infty}\infty$ $\frac\infty{-\infty}$ $\frac{-\infty}{-\infty}$ does l'Hôpital's rule also apply?
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2answers
21 views

Convex function and second devirative

I would like to ask a question about the condition of a convex function. We know that a function $f(x)$ is convex if and only if $f''(x) \geq 0$. But what if a function has more than one variable? ...
2
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3answers
37 views

Interpretation of the curl of a vector field

Let us assume the curl of a vector field is $$ P=(xy)(a_x)+ (y z) (a_y) +(z x) (a_z) $$ Where $ a_x, a_y, a_z $ are unit vectors along x y and z . Then is the curl at a point in the field the ...
2
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0answers
27 views

find the total differential of this equation $ xyz + \sqrt{ x^2 + y^2 + z^2} = \sqrt 2 $

How to calculate the total differential of $ z= z(x,y)$, which is $ xyz + \sqrt{ x^2 + y^2 + z^2} = \sqrt 2 $ at point (1, 0, -1)? The evaluation of mine seems wrong, $ dz= \frac{\partial ...
6
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1answer
43 views

For nonnegative continuous $f$, if $f'(x)-f(x)\leq 0, \forall x\geq 0$ and $f(0)=0$, find the value of $f(1)$. [duplicate]

Let $f(x)$ be a non-negative continuous function such that $f'(x)-f(x)\leq 0, \forall x\geq 0$ and $f(0)=0,$find the value of $f(1)$. $f'(x)-f(x)\leq 0$$\Rightarrow f'(x)\leq f(x)$$\Rightarrow ...
1
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1answer
35 views

Tough problem on sum of infinite series

I've been working on the problem for quite a while but have no idea how to approach it. This proposition arises from a practical probabilistic bound problem, but it seems very deep. Lots of thanks to ...
2
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1answer
30 views

Evaluating a limit I

Consider the limit \begin{align} \lim_{x \to \infty} \left[ \frac{(x+a)^{x+1}}{(x+b)^{x}} - \frac{(x+a-n)^{x+1-n}}{(x+b-n)^{x-n}} \right]. \end{align} It is speculated that the resulting value is ...
2
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4answers
63 views

Does $\displaystyle\sum^{\infty}_{n=1}\left(\frac{n!}{n^n}\right)$ converge or diverge? [duplicate]

Does $\displaystyle\sum^{\infty}_{n=1}\left(\frac{n!}{n^n}\right)$ converge or diverge? I've tried the ratio test, but i'm unsure if I can continue this way. ...
1
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1answer
10 views

Anti-deriving composition of a non-linear activation function on Fourier series?

My pea-brain is not commensurate with the big words in the title. But I'm working on a project where I need to compute definite integrals of the composition $f(g(x))$, where $f(x)$ is any non-linear ...
1
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2answers
28 views

Minimize the area of a wire divided into a circle and square.

A wire is divided into two parts. One part is shaped into a square, and the other part is shaped into a circle. Let r be the ratio of the circumference of the circle to the perimeter of the square ...
3
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1answer
74 views

Integral $\int_0^\infty\text{Li}_2\left(e^{-\pi x}\right)\arctan x\,dx$

Please help me to evaluate this integral in a closed form: $$I=\int_0^\infty\text{Li}_2\left(e^{-\pi x}\right)\arctan x\,dx$$ Using integration by parts I found that it could be expressed through ...
0
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0answers
33 views

Evaluating triple integrals that are bounded

I'm slowly learning how to bound triple integration problems, but this one has me a little confused. $\iiint_D(x+2y)dV$, where D is bounded by the parabolic cylinder, $y = x^2$, and the planes x=z, ...
2
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1answer
28 views

Minimize distance between polynomials, of a certain form, with Laguerre polynomials

A typical problem that I may encounter on an upcoming test looks like this: Find the polynomial $P(x)$ of a degree less than or equal to three that minimizes $$\int_0^\infty (x^4 - ...
0
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0answers
18 views

Closed form solution (formula) for possible events

Let's have 100 time units and 4 possible events A1, A2, B1, B2 that might occur within the 100 units. A1 always occurs before A2, B1 always occurs before B2, t1 < t2 < t3 < t4. There are 2 ...
1
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1answer
32 views

Finding the volume of a solid region

I'm trying to find the volume of the solid region inside the sphere $x^2+y^2+z^2=4$, and the upper nappe of the cone $z^2=3x^2+3y^2$ (I only have to set up the triple integral itself, not evaluate ...
2
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1answer
52 views

Difference between line integrals in complex analysis and real analysis,

The formula in complex analysis is $$\int f(\gamma(t))\cdot(\gamma'(t)dt$$ and the formula in the real variable setting, for a gradient field, is: $$\int F\cdot dr$$ $$=\int f_x\,dx + f_y\,dy + ...
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1answer
41 views

Calculating the limit of a quotient with exponential functions using exponent rules

I need to calculate the following limit $$\lim_{n \to \infty}\frac{3\cdot2^n - 2\cdot3^n}{ 5\cdot2^n - 6\cdot3^n}.$$ Any way, back to our topic, according to my book and wolframalpha, the answer is ...
1
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2answers
49 views

What is a real world example of “zero work” done by a conservative vector field?

I have only a high school physics background, so when I study the later parts of multivariable calculus, e.g., Greens, Gauss, and Stokes' theorems, there are some topics that I only know the ...
6
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0answers
97 views

Wanted: Simple integration theory

Supposing we want to formulate a very primitive theory of integration, the only requirement being that all continuous functions $[a, b]\longrightarrow\mathbb{R}$ be integrable. What is the simplest ...
2
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2answers
46 views

Evaluate $\oint_{C} e^{-x} \sin y \;dx+e^{-x} \cos y\;dy$

I need to evaluate the following integral using Green's theorem $$\oint_{C} e^{-x} \sin y \;dx+e^{-x} \cos y\;dy$$ $C$: from point $E \to F\to G\to H$ ...
0
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1answer
48 views

Why is the discriminant of the discriminant negative?

On this link is a question about functions. My question is, in that question itself, a pivotal part of the solution is to realise that the discriminant of the (positive) discriminant is negative. ...
2
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2answers
77 views

Find the limit of an infinite series

My intuition was to try and see if the series is a Riemann Sum of a function and then see what happens but I can't really see which function fits here. Thanks!
1
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1answer
27 views

confusion about change of variable

If you are integrating $f(x,y)$ over a region and you do a change of variable to $f(u,v)$. The jacobian gives $dx\,dy = du\,dv (\partial x/ \partial u\ \partial y/\partial v - \partial x/\partial v\ ...
0
votes
1answer
31 views

Integral upper bound

Let $A$ be a measurable set and $f$ an integrable function onto $[0,100]$ for example. Having knowledge of the value $\frac{\int_A f d\mu}{\mu(A)}$ (which in some sense is the average value of $f$) I ...
0
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0answers
31 views

When is using the gradient to calculate distance not accurate?

I've read that if you have a function like $y=f(x)$ or $z=f(x,y)$, that you can get the distance from a point $P$ to the closest point on the function by using the gradient. Specifically, you plug ...
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4answers
50 views

Trigonometric substitution and triangles

I'm learning trigonometric substitutions and am having a bit of trouble understanding the intuition behind the conversions (why do most use secant?). If you could explain the conversions geometrically ...
11
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2answers
300 views

Find a closed form of the power series

Let a power series $$S(x)=\sum_{n=1}^{\infty}\frac{x^{n}}{4n+1},$$ then $1$ is the radius of convergence of $S$ .In fact $S(x)$ convergens for each $x\in[-1,1).$ My work is to find a closed form of ...
1
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2answers
70 views

Convergence, Integrals, and Limits question

Let $f: [0,\infty)\to \Bbb R$ be a positive,decreasing monotonic function. Prove the following statement for every a>0 providing the integral on the right side converges. First I managed to ...
1
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2answers
29 views

If $f$ is Lipschitz continuous on a closed interval $[a,b]$ such that $f([a,b])\subseteq [a,b]$ then it has a unique fixed-point

I am stucked at this problem: Prove or give a counter-example for the following sentence: If $f:[a,b]\to\Bbb{R}$ is Lipschitz continuous on a closed interval $[a,b]$ and $f([a,b])\subseteq [a,b]$ ...
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votes
4answers
73 views

The sum of the series $\sum_{n=1}^{\infty}\sin^n(k)$

What is $$\sum\limits_{n=1}^{\infty}\sin^n(k)?$$ Can you find what is the sum of that series. It is convergent not divergent. What if $k=\frac{\pi}{6}$?
1
vote
1answer
27 views

Expected Value of Two Random Variables

X is a random variable with a probability density function $f(x)$, g(x,y) is a function of two variables one of them is the random variable. I have \begin{equation} \int_{-\infty}^{\infty} ...
4
votes
3answers
52 views

Derivative of $f(t)=\frac {1}{\rho}\log (1+\rho t)$

Could you have me to find the ferivative of $$f(t)=\frac {1}{\rho}\log (1+\rho t)$$ with repsect to $t$? And Is it $$\lim_{t\to \infty} \frac {f'(t)}{t}=1?$$ Update: Based on Hint of Surb: ...