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Problem 237 "Mathematical Quickies:270 Stimulating Problems with Solutions" Particle Movement

Here’s another rigorous explanation that doesn’t require thinking about triangles. Assume that the acceleration is at less than $4$ always. Repeatedly integrating and using that $f’(0)=0$, we have ...
Eric's user avatar
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3 votes

What is the limit of the alternating series $F(z)=\sum_{n=1}^\infty(-1)^n z^{T_n}$ as $z\to1$ for a sequence $T_n\sim cn$?

The answer is no. For example, $$ \sum_{n=1}^\infty (-1)^n z^{4n+(-1)^n} = - \sum_{m=1}^\infty z^{4(2m-1)-1} + \sum_{m=1}^\infty z^{4(2m)+1} = -\frac{z^3}{1-z^8} + \frac{z^9}{1-z^8} = -\frac{z^3+z^5+z^...
Greg Martin's user avatar
  • 81.9k
1 vote
Accepted

Do three points of inequality between convex functions imply inequality over an interval?

Let $f(x)=x^2,g(x)=x^4+1/5$. Both are convex functions. We have $f(4/5)>g(4/5)$, $f(3/5)>g(3/5)$ and $f(-3/5)>g(-3/5)$ but $f(0)<g(0)$.
Lucenaposition's user avatar
0 votes

Estimation of the the maximum relative error

Your approach under Edit is correct. It is good to remember, as you did, that the error can be negative. When you see $\rho(V)-\rho(l)$ it can get as large as $\pm 0.5\%$
Ross Millikan's user avatar
3 votes

Determining if a series converges or or diverges

This is perhaps overkill but I think the maths is cute and it doesn't require any appeals to Stirling's approximation etc: For $x = 4$, $\displaystyle\sum\limits_{n=1}^{\infty} \frac{(n!)^2}{(2n)!}2^{...
Adam Dougall's user avatar
2 votes

Determining if a series converges or or diverges

Stirling's approximation (good for intuition, even if it's not acceptable for an explanation) gives you $$ \frac{4^n(n!)^2}{(2n)!} \approx \frac{4^n(\sqrt{2\pi n} n^n e^{-n})^2} {\sqrt{2\pi(2n)} (...
Matthew Leingang's user avatar
1 vote

Prove $\frac{d}{dx} \frac{f(x)-f(a)}{x-a} \geq 0$ if $f(x)$ is convex without twice differentiability.

I don't think you're proving what you were asked. You said that you were asked to show that $f'(x)$ is increasing but instead showed that $\frac{f(x)-f(a)}{x-a}$ is, and $f'(x)$ and the other ...
David G's user avatar
  • 133
2 votes

Determining if a series converges or or diverges

For $x = -4$, think about the alternating series test. You have to check that the terms are indeed decreasing to zero in absolute value. For $x = 4$, think about Stirling's approximation to find an ...
Fançois Gatine's user avatar
0 votes

usage of Leibniz notation for things like $\frac{d^2y}{dt^2}$ and $\frac{dy'}{dy}$

For higher-order differentials to work algebraically, you need to adopt a notation that is a little non-standard. The typical notation, $\frac{d^2y}{dx^2}$ does not allow for algebraic manipulations. ...
johnnyb's user avatar
  • 3,589
1 vote

Double integral of the form exp(-a(x-y)^2)

I'm gonna sketch out the solution, the questions are welcome in the comments: First, apply the change of variables in the following form: $$ \left\{ \begin{array}{} u&=& x-y \\ v&=& x+...
Egor Larionov's user avatar
4 votes
Accepted

Calculus: Is $(x-1)/f(x)$ decreasing?

To summarize the discussion in the comments: The claim is false. A simple counterexample is given by $$f(x)=(e^{-x}+1)(x-1)$$ The argument the OP invokes in the differentiable case is likewise false (...
lulu's user avatar
  • 71.8k
0 votes

Does the value of a convergent series equal the sum of its non-negative terms minus the sum of the absolute values of its negative terms?

Yes it is true, and more elementary than the two theorems you are quoting: Assuming $\sum a_n$ is absolutely convergent, simply write $$\sum_{n\le N}a_n=\sum_{n\le N\atop a_n\ge0}a_n-\sum_{n\le N\atop ...
Anne Bauval's user avatar
  • 39.5k
0 votes

Must the $x$ and $y$ axes have the same units? (Coordinate Geomtry)

Take South St. Paul, Minnesota, USA. House numbers are 1600 per mile along one axis, and 800 per mile along the other, to fit their particular block structure. I.e.. nobody is forcing one to use the ...
Dan Jacobson's user avatar
0 votes

Is there such a thing as the "Second Passage Time"?

Based on the comments provided by @Zack Fisher, I tried to write an answer: Using the Law of Conditional Probability: If $ X $ and $ Y $ are two random variables, the joint probability density ...
konofoso's user avatar
  • 715
0 votes

How to calculate the functional derivative of this composite function?

Use chain rules and commutativity $$\delta \int = \int \delta$$ $$\frac{\delta }{\delta u} \log f(g(u))= \frac{f'(g(u))}{f(g(u))} \ \frac{\delta}{\delta u} g(u)$$ with $$ \frac{\delta}{\delta u} \ g(u)...
Roland F's user avatar
  • 3,150
1 vote

Does there exist a positive sequence with these two properties?

While the answer by @John B is of course correct, I wanted to give a simple explicit construction of such a sequence. Ignoring the $(-1)^n$ for a moment, you essentially want multiples of $\sqrt2$ ...
anankElpis's user avatar
  • 1,403
2 votes

Does there exist a positive sequence with these two properties?

Let's forget first about $(-1)^n$. Given $n\in\mathbb N$, take $c_n>0$ such that $|\{c_n\}-1|<\frac1n$. Of course, $$ |\{m+c_n\}-1|=|\{c_n\}-1|<\frac1n\tag1 $$ for any positive integer $m>...
John B's user avatar
  • 17.2k
-1 votes

If $x$ and $y$ are arbitrary real numbers with $x < y$, prove that there exists at least one real satisfying $x < z < y$

There is a much simpler way of proving this, rather than using sup and inf... Let $x$ and $y$ be arbitrary real numbers with $x<y$. Let $z=\frac{x+y}{2}$. I will show that $x<z$. $z-x=\frac{x+y}...
mitcheljh's user avatar
0 votes

Partial Derivative of a Scalar Function

$ \def\bR#1{\Big[#1\Big]} \def\BR#1{\left[#1\right]} \def\lR#1{\Big(#1\Big)} \def\LR#1{\left(#1\right)} \def\frob#1{\left\| #1 \right\|} \def\q{\quad} \def\qq{\qquad} \def\qif{\q\iff\q} \def\qiq{\...
greg's user avatar
  • 37k
2 votes
Accepted

Integrate the product of a heaviside step and the absolute value?

I like the direction that you're heading. When integrating Heaviside functions and/or absolute values, I recommend breaking up the integration domain into regions where they are constant and linear ...
Harrison Eggers's user avatar
1 vote

Understanding epsilon-delta proof regarding dense sets

As said in comments it is good to give an example. As you have guessed, $h(x^{\star})$ may go to $0$ as $x^\star$ goes to $x_0$. You can easily construct the examples. Consider $g(x) = 2x^2$ and $h(x) ...
Yathi's user avatar
  • 2,470
0 votes

Evaluating $\lim_{x\to\infty}\left[\cos\frac1x\right]^{h(x)}$, where $h(x)=\frac{x^4+x^2-1}{2x+1}\sin\frac1x$

I found this cool inequality and never managed to used it before, so here it goes: $x \in (0, \frac{\pi}2) \implies \frac{\sin(x)}{x} > \frac 2{\pi}$ of course for $x > \frac 2{\pi}$ we will ...
hellofriends's user avatar
  • 1,970
1 vote

Using Graph of $\frac{1}{x}$ to Find $\delta$

The solution to this specific question was $4-\frac{1}{.35}$. Formatting was very specific for the website I was using. To get this answer, I set $f(x)$ equal to the points on the Y-axis, factored, ...
Squishy698's user avatar
2 votes

Evaluating $\lim_{x\to\infty}\left[\cos\frac1x\right]^{h(x)}$, where $h(x)=\frac{x^4+x^2-1}{2x+1}\sin\frac1x$

By substituting $t=1/x$ we get $t\to 0^+$ and $$h(1/t)\log \cos t={1+t^2-t^4\over 2t^3+t^4}\sin t\log \cos t \ (*)\\ ={1+t^2-t^4\over 2+t}{\sin t\over t}\,{\cos t -1\over t^2}\,{\log \cos t\over \cos ...
Ryszard Szwarc's user avatar
0 votes

Using Graph of $\frac{1}{x}$ to Find $\delta$

Hint: First observe that solving for $1/x = .35$ and subtracting from $4$ will work but not solving for $1/x = .15$ and so on. And also check on which side you have to round. Edit: you can also check ...
Sounak's user avatar
  • 41
8 votes

Evaluating $\lim_{x\to\infty}\left[\cos\frac1x\right]^{h(x)}$, where $h(x)=\frac{x^4+x^2-1}{2x+1}\sin\frac1x$

$\textbf{Hint:}$ For large $x$, we have that $$\cos\left(\frac{1}{x}\right) \sim 1 - \frac{1}{2x^2}$$ and $$\frac{x^4+x^2-1}{2x}\sin\left(\frac{1}{x}\right) \sim \frac{x^2}{2}$$ therefore your limit ...
Ninad Munshi's user avatar
  • 35.9k
2 votes
Accepted

Bump function with integral $1$ and value $1$ at zero

Consider $$g_0(x)=f\left(x-\frac12\right)f\left(\frac12-x\right)$$ This is a smooth bump function centred at the origin, but $g_0(0)\neq1$. We can fix that with $g_1(x)=\frac{g_0(x)}{g_0(0)}$. Now for ...
Arthur's user avatar
  • 201k
2 votes

Problem 237 "Mathematical Quickies:270 Stimulating Problems with Solutions" Particle Movement

Here's a more mathematically rigorous explanation of what the solution is saying: Suppose there exists a path $f(t)$ such that $f'(t)$ does not exceed the bounds of this isosceles triangle (whose path ...
Sharky Kesa's user avatar
  • 4,402
1 vote

Integration of $1/x$ as a Riemann sum

$$\lim_{t\to 0}\frac{a^t - 1}{t} = \ln{a}$$ so $$\lim_{n\to \infty}\frac{(b/a)^{1/n} - 1}{1/n} = \ln{b/a}$$
user23492240's user avatar
4 votes
Accepted

Why can the Binomial Distribution be Approximated by a Normal Distribtuion?

For the second factor in step 8, let $v=1/n$ , and take logarithm to get $$ \frac{1}{v} \log\left\lbrace 1 + u(v)\right\rbrace, $$ where $$ u(v)=p\left( \exp\left\lbrace \frac{\sqrt{v}t}{\sqrt{p(1-p)}}...
Zack Fisher's user avatar
3 votes
Accepted

Recursive piecewise integral formula

When $n = 1$, you do not use the recursion formula. Instead, you just observe $$I_1 = \int \frac{dx}{x^2 + a^2} = \frac{1}{a} \arctan \frac{x}{a} + C.$$ When $n = 2$, the recursion formula yields $$\...
heropup's user avatar
  • 141k
0 votes

Issue in numerical integration of $\frac{1}{4\pi^{3/2}i}\int_Ce^{-\frac{z^3}{12}-tz}z^{-1/2}dz$

$$\operatorname{Ai}^2(t)=\frac{1}{{4 \pi ^{3/2} i}}{\int_{\sigma-i \infty }^{\sigma+i \infty } e^ \left(\color{red}+\frac{z^3}{12}-t z\right)\cdot z^{-\frac{1}{2}} \, dz}$$ Use SciPy Gaussian ...
gpmath's user avatar
  • 1,091
2 votes

Why can the Binomial Distribution be Approximated by a Normal Distribtuion?

Consider the Binomial PMF: $$ P(X=k)=\binom{n}{k} p^k(1-p)^{n-k} $$ We will use Stirling's Approximation: $$ n!\approx \sqrt{2 \pi n}\left(\frac{n}{e}\right)^n $$ ...for the binomial coefficient: $$ \...
vallev's user avatar
  • 396
3 votes

What is the sign of $I_n = \int _{0}^{1}\frac{x^{2n+1}}{x^{2}+1}dx$

Trivially, ${x}^{2 n + 1} > 0$ for all $x \in \left( 0, 1 \right]$, and $1 + {x}^{2} > 0$, so the integrand is decidedly positive for all $x \in \left( 0, 1 \right]$. Trivially, therefore, the ...
Simon's user avatar
  • 1,451
5 votes
Accepted

What is the sign of $I_n = \int _{0}^{1}\frac{x^{2n+1}}{x^{2}+1}dx$

A recursive formula holds: $$ \begin{align} I_n = \int_0^1 \frac{x^{2n+1}}{x^2+1} dx &= \int_0^1 \left(\frac{x^2} {x^2+1}\right) x^{2n-1} dx\\ &= \int_0^1 \left(1-\frac{1}{x^2+1}\right)x^{2n-1}...
zetko's user avatar
  • 319
1 vote

Find a sequence function for combinatorial sequences

Given by Mathematica, the sequence is $$a_{n}=\frac{30 (n-3)\, a_{n-1} +\left(18370 n^2-82832 n+92261\right)}{30 n+17}\qquad \text{with} \qquad a_1=0$$.
Claude Leibovici's user avatar
0 votes

Find minimum value of $f(x)= x^{1.5} + x^{-1.5} -4(x + x^{-1})$

Another way without using derivatives. I will use the sum of cubes formula and the completing the square method. $\begin{align}f(x)&=x^{1.5}+x^{-1.5}-4(x+x^{-1})=\\[3pt]&=\left(\sqrt x\right)^...
Angelo's user avatar
  • 12.4k
3 votes

Why this formula cannot be used here?

When the integrand is also a function of the independent variable, the formula takes an extra term. $$\frac{d}{dx}\int_{u(x)}^{v(x)}f(t, x)\,dt=f(v(x))\frac{dv(x)}{dx}-f(u(x))\frac{du(x)}{dx}+\int_{u(...
Yves Daoust's user avatar
2 votes

Why this formula cannot be used here?

In the formula, the function to be integrate only depends on $t$, while yours depends on both $t$ and $x$. In this case you can easily move forward just by noting that $e^{x-t} = e^x e^{-t}$: $$ g'(x)=...
PierreCarre's user avatar
  • 21.6k
1 vote

Why $\lim_{x \rightarrow \infty } \frac {P(x)}{Q(e^x)} = 0$ for polynomials $P(x)$ and $Q(x)$?

We will use two properties. if $\deg R<\deg S$ for two polynomials $R(x), S(x),$ then ${R(x)\over S(x)}\underset{x\to\infty}{\longrightarrow} 0.$ $e^t>t>0$ which implies $e^x=(e^{x/m})^m> ...
Ryszard Szwarc's user avatar
1 vote
Accepted

Why $\lim_{x \rightarrow \infty } \frac {P(x)}{Q(e^x)} = 0$ for polynomials $P(x)$ and $Q(x)$?

I'm not sure I understand your objection. In general, if the numerator of a fraction grows sufficiently slower than its denominator, regardless of which infinity that denominator goes to ($+\infty$ or ...
PrincessEev's user avatar
  • 45.8k
5 votes

Why $\lim_{x \rightarrow \infty } \frac {P(x)}{Q(e^x)} = 0$ for polynomials $P(x)$ and $Q(x)$?

If $Q(x)=a_0+a_1x+\cdots+a_nx^n$, with $n\in\{1,2,3,\ldots\}$ and $a_n\ne0$, then\begin{align}\lim_{n\to\infty}\frac{Q(e^x)}{e^{nx}}&=\lim_{n\to\infty}a_0e^{-nx}+a_1e^{-(n-1)x}+\cdots+a_{n-1}e^{-x}...
José Carlos Santos's user avatar
1 vote
Accepted

Prove that a functional equation has at least one root

First of all we need to define $g\circ$$f:[0,1]\rightarrow[-1,4]$ where $g(x)$=$x^3$+$\beta$$x^2$+$\gamma$$x$ just as you did. Now we can apply the Intermediate Value Theorem since the defined ...
Elfryionnn's user avatar
1 vote

Proving that $\lim_{h\to 0 } \frac{b^{h}-1}{h} = \ln{b}$

$$\lim_{h\to 0}{b^h-1\over h}$$ First, we define a variable t such that $b^h-1=t$. Therefore, $b^h=t+1$ and $h=\log_b{(t+1)}$. It should be noted that, if we take the limit as $h$ goes to $0$, $t$ ...
Eli Berk's user avatar
0 votes

Squaring a function, getting a surprising result

From the previous comments, it seems that you want to find the minimum distance between the origin and the parabola $y=x^2-2x+5$. You are correct that the distance equation in this case is $d=\sqrt{x^...
VV_721's user avatar
  • 375
1 vote

Number of times a continuous function changes sign in an interval

Define the operator $T$ which associates to a function $f$, the funcn+1ion $Tf$ such that $Tf(x)= f_1(x) = \int_0^x f(t) dt$. Then $f_n$ is none other that $T^n f$. One can see it by induction: if $...
alex440's user avatar
  • 531
1 vote
Accepted

Number of times a continuous function changes sign in an interval

For every $0<x\leq a$ and $n\in\mathbb{N}\cup\{0\}$, the factor $t\mapsto (x-t)^n \mathbb{1}_{[0,x]}$ is positive in $(0,x)$. For $n\geq 1$, integrating by parts gives $$ \begin{align} f_{n+1}(x)&...
Boris PerezPrado's user avatar
4 votes

Find solution of the IVP $ y' = y+ \frac12 |\sin(y^2)|,\,\, x>0,\,\, y(0) = -1$

You have $0 \leqslant y' - y \leqslant \frac{1}{2}$. Multiplying by $e^{-x}$ yields $0 \leqslant y'(x)e^{-x} - y(x)e^{-x} \leqslant \frac{1}{2}e^{-x}$. Integrating on $[0,x]$, after noticing that $y'(...
Didier's user avatar
  • 19.6k
1 vote

About solution to homogeneous ODE $u'' + u = 0$

Another approach, let $f(x) = u(x)^2+(u'(x))^2$. Then $f(0) = 0$ and $f'(x) = 0$ hence $f(x) = 0$.
copper.hat's user avatar
  • 174k
2 votes

Find solution of the IVP $ y' = y+ \frac12 |\sin(y^2)|,\,\, x>0,\,\, y(0) = -1$

You should check first that the right hand side is locally Lipschitz in $y$. Picard-Lindelöf implies existence and uniqueness. Under these circumstances, solutions to first order autonomous ODEs (like ...
Hyperbolic PDE friend's user avatar

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