**82.604 calculus questions.**

For which $p>0$ does $\sum_{n=3}^{\infty}\frac{\log(n)}{n^p}$ converge? I tried all the criteria for series convergence I know, but I'm not getting any further with this exercise. I'm not asking ...

$$x_{n+1} = x_n + \sin x_n$$
$$x_{n+1} = \sin \left(\frac {\pi} {2} x_n\right)$$
How to solve these? Or, at least, what can be said about their behavior and limits?

Suppose we are given a continuous function $F$ which is also increasing and bounded. I was reading a proof which involved asserting that
$\lim_{n \to \infty}\frac{1}{n}\int^{b+\frac{1}{n}}_{b} F(x)...

Hello I am currently trying to solve a differential equation using Matlab; however, my problem is not with the code but the math. I have achieved at an answer to this question 252.0178 m/s. That is ...

I think it is homogeneous of order one, but the book says it is homogeneous of order two. I just want a confirmation. Let $F(a_1,a_2,...,a_n) = \left\{\sum_{k=1}^n a_k^{1/2}\right\}^2$, then
$$F(wa_1,...

(The Cauchy-Schwarz Master class by Michael Steele, page 121) It is given that $\ln(1+x) = x+O(x^2)$ as $x\to 0$. Then, as $t\to 0$ one has
$$\frac{1}{t}\ln\left\{ 1+ t\sum_{k=1}^n p_k\ln x_k + O(t^2)...

Find the limit below
\begin{align} \lim_{r \rightarrow 0}\frac{-r^2}{2 \left(\sqrt{1-\frac{r^2}{4}}-1 \right)}\end{align}
I look at the graph and see that it seems to be going to zero. This makes ...

For all the primes smaller than 100, find the numbers that have a digit sum that is equal to their digit product. The solution to the challenge is the sum of all the resulting numbers.
A prime is a ...

I believe these are the first steps:
$$
f´(x)=2xe^{-x}-x^2 e^x$$
$$f´(a)= 2ae^{-a}-a^2 e^a
$$
Since it goes through the origin, the tangent line on $a$ is given by:
$$y-0 = f´(a)(x-0)...$$ but I don´t ...

When I learn real analysis, I encountered $\int_0^1 \frac{x^2-y^2}{(x^2+y^2)^2}dy$, I don't know how to compute that, although I've tried to make $y = \tan\theta$ but I feel it's useless.

$$\lim_{(x,y) \to (0,0)} \frac{\sqrt{1+x^2+y^2}-1}{\arcsin{\left(x^2+y^2+x \right) }-\arcsin{x}}=0$$
I have tried using polar coordinates, $x = r\cos\theta$, $x = r\sin\theta$ and I do not get ...

Any ideas/hints on how to construct a non-decreasing function on $[0,1]$ whose set of discontinuities is not closed?
The motivation is that I noticed that most "regular" functions have closed (...

Evaluate$$
\int_0^{\infty}{\frac{\arctan x\operatorname{Li}_2(-x)}{x^2}dx}$$
My attempt:
By Integrating by parts I got
$$
I=0+\int_0^{\infty}{\frac{\ln \left( x+1 \right)}{x}\left( \ln x-\frac{1}{2}\...

Let $g: \mathbb{R} \rightarrow \mathbb{R}$ be a monotonic function that is smooth. I am considering the quotient $\frac{g(f(x_1))}{g(f(x_2))}$ for some $x_1,x_2 \in \mathbb{R}$. Is is it possible to ...

1) I am asked to find the radius of convergence using the root criterion of:
$$\sum_{n=1}^{\infty}[(p (p + 1) (p + 2) ... (p + (n-1))) / n!] z ^ n$$
where $p\in\mathbb{N}$
Well my idea here was to ...

I'm having trouble determining the convergence of the series:
$$
\sum_{n=1}^{\infty}\left[1-\cos\left(1 \over n\right)\right].
$$
I have tried the root test:
$$\lim_{n\rightarrow\infty}\sqrt[n]{1-\...

in my notebook there is a statement that "we called non real numbers imaginary numbers such as $\sqrt {-2}$ " i needed to know that is true or false.
1. do we have non real numbers beyond complex ...

Consider the following series for $x \in [0,1)$:
$$
\sum \frac{x^n}{1+x^n}
$$
I figured that it converges, by using the ratio test:
$$
\left| \frac{a_{n+1}}{a_n} \right| = \frac{x^{n+1}}{1+x^{n+1}} \...

Given that
$ \prod\limits_{n=2}^{\infty}{e(1-\frac{1}{n^2})^{n^2}}$
I want to calculate this infinite product. Your helps will be appreciated.

Inspired by L'Hosopital I started considering “ L'Hospital's equations ”
Let $x$ be real and for all $ x > s $ we have $f(x),g(x)$ map to the reals.
Let $*^{(n)} $ denote the $n$ th derivative. $...

This question concerns an interesting proof of the fact that $\sin^2(x) + \cos^2(x) = 1$, but only using the series that defines them, not any trigonometry. So define
$$
s(x) = x - \frac{x^3}{3!} + \...

I have come across the term 'functional'. How is a 'functional' different from a 'function'? The exact term I came across was 'statistical functional.'
In terms of the background, can you please ...

Let $f(x) =x^3+3x+1$. Find area bounded by $y=f^{-1}(x)$, the ordinates $x=-3$,$x=5$ and X-axis.
My Attempt:
The required area will be equal to area enclosed by $y=f(x)$,the y-axis between the ...

I found this problem in some notes linked on the course page of a particular online class that I am taking. I am not sure if I am allowed to provide them here, so hopefully context does not matter too ...

Check if the following series converge or diverge:$\sum_\limits{n=1}^{\infty}(1-\cos(\frac{\pi}{n}))$
I have tried the integral test since the series are decreasing to zero as $n\to\infty$, but $\...

10.22 Let $F(y)$ be defined by $$F(y)=\int_0^\infty e^{-x^2}\cos(2xy) \,\mathrm dx$$ for $y\in\Bbb R$. Show that $F$ satisfies the differential equation $$F'(y)+2yF(y)=0$$
and deduce that $F(y)=\...

I am looking for a good online video resource to start studying Calculus. I am studying it alone, not part of any school or university. Trying to learn and enhance my mathematical skills.
Thanks!

How to calculate this limit?
$$\lim_{x\to 0}\frac{\sin^2\left(3x\right)}{\sin^3\left(\ln\left(2x+1\right)\right)}$$
Applying L'Hopitals rule seems to be too complex.

I am trying to find maximize $9x+4y$ subject to the condition $|3x|+|2y|\le 1$. In this problem, have to use the Lagrange multiplier method, but the function mod $3x+\operatorname{mod} 2y$ is not ...

I have an exam next week and I'm not going to be able to bring any piece of paper with me. I need to memorize these theorems, both for improper integrals and for function series. How do I go about ...

For my homework I have to take the path integral of $F=<\arccos x,xy-e^y>$ over the triangular path from $(0,0)$ to $(2,3)$ to $(2,0)$ and back to $(0,0)$. I have broken this integral into three ...

I am working on the question below and I am getting stuck.
Consider the surge function $y=axe^{-bx}$ with $a$ and $b$ positive constants.
(a) Find the local maxima, local minima, and ...

I'm supposed to use the product rule to differentiate:
$$
\frac{d}{dx} (3x + 1)^{\frac{3}{2}}(2x + 4).
$$
This gives me:
$$
2(3x+1)^{\frac{3}{2}} + \frac{9}{2}(2x + 4)(3x + 1)^{\frac{1}{2}}.
$$
My ...

A function of degree 4 has the graph as shown.
How many solutions of the equation $[f'(x)]^2=f(x)\cdot f''(x)$.
The answer is $0$ solution. I have tried to express as:
$$\int\frac{f'(x)}{f(x)}dx=\int\...

$Y(t)=a[S(t)+N(t)]^2$,$S(t)$ and $N(t)$ are both Gaussian random process and WSS with zero mean,and $S(t)$ is independent of $N(t)$
\begin{align}
R_Y(t_1,t_2) & =E[Y(t_1)Y^*(t_2)] \\
& =a^2E[...

Hi I'm looking for some help with the function
$$f(s,t)=a(2a-1)|t-s|^{2a-2}.$$
I'm trying to solve the following definite integral
$$\int_0^t\int_0^sf(u,v)dudv=a(2a-1)\int_0^t\int_0^s|u-v|^{2a-2}dudv....

I try to solve following inequailty $-3t^4-4Bt^3-2B^2t^2+(6D-2BC)t+2BD-C^2 \leq 0$ where $B$, $C$ and $D$ are real numbers.
Say $g(t)=-3t^4-4Bt^3-2B^2t^2+(6D-2BC)t+2BD-C^2$.
We observe that $g''(t)=-...

Let $A$ be a domain in $\mathbb R^2$ whose boundary $\gamma $ is a smooth positively oriented curve and whose area is $|A|$.
Find a function $F:\mathbb R^2\to \mathbb R$ such that $$\frac{1}{|A|}\...

How can this limit be evaluated without the use of L'Hopital's rule. I already understand how to evaluate it with the use of it.
\begin{equation}
\lim_{x\to 0} (\frac{e^{3x}-1}{x})
\end{equation}
I ...

For which $\alpha > 0 $ does the series
$$ \sum_{n=1}^{\infty} \arctan(\frac{1}{1 + n + n^{\alpha}}) $$
converge? I suspect it does for $\alpha > 1$ but how do I show that and that it doesn't ...

We have the following situation
The goal is to find $$\lim_{a \to b } \frac{a-b}{c-d} $$
Thought
As $a $ tends to $b$ then we see we are gonna have a rectangle which means that $2 \alpha $ is gonna ...

$f(x) = ax^3+bx^2+cx +d$, determine a, b, c, and d such that the graph of $f$ has a extreme in $(0,3)$ and a point of inflection in $(-1,1)$.
When is a quadratic I know that the formula $V=(\frac{-b}{...

I want to calcurate
$ \lim_{n \to \infty} \int_{(0,1)^n} \frac{n}{x_1 + \cdots + x_n} dx_1 \cdots dx_n $
I met this in studying Lebesgue integral. But, I don't know how to do at all. I would really ...

The positive rational numbers may be arranged countably as the series $$1/1, 2/1, 1/2, 3/1, 2/2, 1/3, 4/1, 3/2, 2/3, 1/4,\dots$$
Show that $p/q$ is the $$[1/2(p+q-1)(p+q-2)+q]^{\text{th}},$$ term of ...

I have an integral function describing the toughness of a material.
As the material behaves differently in unloading so that we have different functions to describe the behaviours.
$$f(ε) = σ$$
$$\...

If an object is thrown in the air with an initial position of $(0,0,4)$ in feet and initial velocity $(200,0,256)$ in feet per second, and the acceleration is $(0,8,-32)$. Find the position and ...

Suppose you have a product of three binomials each of which can be represented as binomial series:
$(1-x)^{\frac{1}{2}}\times (1+4x)^{\frac{1}{2}}\times (1-2x)^{-2}$
How do you find the center and ...

I have the following integration problem:
$$ \int_{0.5}^{1} \frac{1}{\sqrt{2x-x^2}} $$
And I can see I should probably be completing the square here. I may be missing something extremely obvious, ...

$f(x) = \frac{1}{x}$
The function is supposedly continuous, but there is an infinite discontinuity at $x = 0$. I'm confused. Isn't the definition of a continuous function as one that doesn't have ...

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