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

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2
votes
3answers
59 views

Prove that a certain sequence is increasing and find its limit: $a_1 = 1$ and $a_{n+1}=n(1+\ln a_n)$ (and $(a_n)^\frac{1}{n}$)

Let $a_1 = 1$ and $$a_{n+1}=n(1+\ln a_n).$$ I have to find its limit. I want to prove that it is increasing for starters, but I'm already stuck. What should I do? I have also to find the limit of ...
0
votes
2answers
82 views

How to prove that the sum of a convergent geometric series of the form $1 + r + r^2 … + r^n > 1/2$?

I am trying to prove that the sum of a convergent geometric series of the form \begin{equation*} 1 + r + r^2 .... + r^n > \frac{1}{2} \end{equation*} but I have no idea how to go about this. ...
0
votes
2answers
20 views

Partition set of $n$ elements until each partition contains $1$ element. Must terminate after exactly $n-1$ iterations?

Suppose I have a set of $n$ elements and I want to partition the set (split into two) until each partition contains a single element. How do I see that the terminating case must occur after exactly ...
0
votes
0answers
14 views

Proving that average value of $u$ around a circle is the value of $u$ at the centre.

I would like to prove that: If $u(x,y)$ is harmonic in a domain containing a disk of radius $r$ with boundary $C_r$ $\implies$ the average value of $u$ around the circle is the value of $u$ at the ...
0
votes
2answers
20 views

Maximizing profit function given cost and demand functions

I am given the demand function $$D(x)=10x^2 + 50x$$ and a total cost of $$C(x) = x^3 + 10x$$ where $x$ is the number of units demanded. I am asked to maximize the profit so what I did is I used the ...
19
votes
1answer
495 views

Is there a function having a limit at every point while being nowhere continuous?

Is there a function $\,f:\mathbb{R}\rightarrow\mathbb{R},\,$ that has a limit at every point but is continuous nowhere?
5
votes
1answer
2k views

Limit of the sequence $\{n^n/n!\}$, is this sequence bounded, convergent and eventually monotonic?

I am trying to check whether or not the sequence $$a_{n} =\left\{\frac{n^n}{n!}\right\}_{n=1}^{\infty}$$ is bounded, convergent and ultimately monotonic (there exists an $N$ such that for all $n\geq ...
1
vote
3answers
2k views

limits of sequences exponential and factorial: $a_n=e^{5\cos((\pi/6)^n)}$ and $a_n=\frac{n!}{n^n}$

Compute the limit $\lim\limits_{n\to\infty}a_n$ for the following sequences: (a) $a_n=e^{5\cos((\pi/6)^n)}$ (b) $a_n=\frac{n!}{n^n}$ For part (a) do I just take the limit of the exponent part and ...
5
votes
3answers
117 views

How can I calculate $\lim_{n \to \infty} (1 + \frac{1}{n!})^n$ and $\lim_{n \to \infty} (1 + \frac{1}{n!})^{n^n}$?

How do you calculate the following limits? $$\lim_{n \to \infty} \left(1 + \frac{1}{n!}\right)^n$$ $$\lim_{n \to \infty} \left(1 + \frac{1}{n!}\right)^{n^n}.$$ I really don't have any clue ...
1
vote
2answers
30 views

Convergence of this alternating series: $\sum_{k=0}^\infty \frac{(-1)^k}{(k+1)C^k} = C \log \frac{C+1}{C}$

I "heard" the following formula for any $C \ge 1$: $\sum\limits_{k=0}^\infty \dfrac{(-1)^k}{(k+1)C^k} = C \log \dfrac{C+1}{C}$ Is it correct? What would be a proof?
2
votes
1answer
182 views

Is Calculus a requirement for Better Statistics?

Is Calculus really required to be better at Statistics and Probability and to be a good Data Science person? Arthur Benjamin Says "Very few people actually use calculus in a conscious, meaningful ...
3
votes
2answers
27 views

Determine whether series converges or diverges

$$\sum_{n=1}^{\infty}\frac{\sin\left(\frac{5\pi}{3}n\right)}{n^{\frac{5\pi}{3}}}$$ Hello, I thought about using Squeeze Theorem but the 5π/3 threw me off. Thanks in advance.
6
votes
5answers
280 views

How is the concept of the limit the foundation of calculus?

My casual study of mathematics and calculus introduced me to the notion that calculus didn't find a firm foundation until Cauchy, Weierstrauss (et al) developed set theory some ~100 years after Newton ...
2
votes
1answer
40 views

Can critical point that $f''$ has different sign in its every neighborhood be a local extreme point?

Suppose that $f$ is a second order derivable function on $[0,1)$, and $f'(0)=0$. It is true that: If there exits $\delta>0$ such that $f''(x)\geq0$ for all $x\in[0,\delta)$, then $0$ is a local ...
2
votes
3answers
78 views

L'Hôpital's rule exercise with natural log function

I'm looking for some advice on the following exercise: $$\lim_{x \to 0^+}{\ln{(\frac{1}{x}})}^x$$ This is my work so far: $$\lim_{x \to 0^+}{\ln{(\frac{1}{x}})}^x = \lim_{x \to ...
2
votes
1answer
36 views

Trying to integrate $\iint_D x^4\tan(x)+3y^2 \,dA$.

I'm trying to integrate $\iint_D x^4\tan(x)+3y^2\, dA$ in domain $D=\{(x, y) \in \Bbb R^2 \mid x^2+y^2\le4, y\ge0\}$. Domain is simple enough; half circle of radius 2 over $x$ axis. Converting to ...
1
vote
1answer
34 views

Explanation of taylor series

I understand that for a Taylor series of a function $f(x)$, centered around the point a, the general expression can be written as: $$ \begin{align} &f(x) \\ &= f(a) + f'(a) (x-a) + ...
-1
votes
3answers
53 views

Area of a region under the mapping $f$

Consider the function $f:\mathbb R^{2} \to \mathbb R^{2}$ given by $f(x,y)=\left(e^{x+y},e^{x-y}\right)$. Area of the image of the region $\{(x,y)\in \mathbb R^{2} | 0<x,y<1\}$ under the mapping ...
3
votes
4answers
99 views

Compute $\lim_{n \to \infty} \left(\frac{1}{\sqrt{n^3+1}} + \frac{1}{\sqrt{n^3+4}} + \cdots + \frac{1}{\sqrt{n^3+n^2}}\right)$

How do I evaluate the following limit? I guess I should do a comparison, but I've got no clue about what to do. Could you give me a hand? $$\lim_{n \to \infty}\left( \frac{1}{\sqrt{n^3+1}} + ...
1
vote
0answers
45 views

Uniformly convergence question

I have the following exercise: Let $\varphi:[0,+\infty)\to \mathbb{R}$ an increasing continuous function that satisfies that $1/2\leq \varphi(x) <1$ for all $x>0$. Let $f_0:[0,+\infty)\to ...
2
votes
1answer
18 views

Upper bound for the infimum of $(K-x)^2 + (T/x)^2$

I have a function $$f(x)=(K-x)^2 + \left( \frac{T}{x} \right)^2$$ where $K$ and $T$ are positive constants and $x>0$. The function $f$ (hopefully) has an infimum, in terms of $K$ and $T$. I ...
3
votes
2answers
67 views

Finding line that divides an area into equal halves.

My question is simple, but I am not getting the answers for some reason. The question is: Consider the area enclosed between the graph of $y = 1 - x^2 $and the $x$ axis. Which line parallel to the ...
0
votes
2answers
43 views

Calculate: $\lim_{n \to \infty} \frac{2^{\sqrt{(\ln n)^2 + \ln n^2}}}{n^2+1}$ and $\lim_{n \to \infty} \frac{10^{\sqrt{(\ln n)^2 + \ln n^2}}}{n^2+1}$

I have to evaluate the following limits (which are similar). However, I don't know how to evaluate them. Could you give me a hand? $$\lim_{n \to \infty} \frac{2^{\sqrt{(\ln n)^2 + \ln ...
0
votes
0answers
23 views

Maximizing the following function

I need to find values of $k_1$, $k_2$ and $k_3$ that maximize $C^{m_1}_{mm_1} \cdot C^{m_2}_{k_1-mm_1} \cdot C^{n_1}_{nn_1} \cdot C^{n_2}_{k_2-nn_1} \cdot C^{p_1}_{pp_1} \cdot C^{p_2}_{k_3-pp_1}$ ...
0
votes
1answer
26 views

Prove family of function is not equicontinuous [on hold]

Assume that $f \in C([0, +\infty))$ is not constant function prove tha family $f(t):=f(nt)$, $n \in \mathbb{N}$ at $(0,1)$ is not equicontinuous.
-4
votes
2answers
63 views

Why is $n^\sqrt n - 2^n \to - \infty$ and $\sqrt n ^n - 2^n \to +\infty$ [on hold]

Can you explain technically why the following limits are correct? $$n^\sqrt n - 2^n \to - \infty$$ and $$\sqrt n ^n - 2^n \to +\infty$$
-1
votes
2answers
51 views

How do I calculate the following limit? $\lim_{n \to \infty} n ((8 + \sin (2^\frac{1}{n}))^\frac{1}{3} -2)$

How do I calculate the following limit? I'm short on ideas for this one: $$\lim_{n \to \infty} n ((8 + \sin 2^\frac{1}{n})^\frac{1}{3} -2)$$
1
vote
0answers
27 views

References for the following functional

In many of the types of problems Ive looked at the following quantity keeps arising and I was wondering if anyone knew any references I could look at to learn some its properties. Take any function ...
3
votes
4answers
77 views

How do I calculate $\lim_{n \to \infty} n^\frac{1}{n} (n+1)^\frac{1}{n+1} … (2n)^\frac{1}{2n}$

How do I calculate the limit of the following sequence? $$\lim_{n \to \infty} n^{\frac{1}{n}} (n+1)^\frac{1}{n+1} ...... (2n)^\frac{1}{2n}$$
0
votes
1answer
30 views

Integral transformation

I'm trying to do a transformation of an integration, I have that $$\int_{0.5}^1\int_0^{0.5}e^{xy}xydxdy$$ And I want to get that integrate $$\int_0^1\int_0^1 f(x,y)dxdy$$ Where the value of the ...
0
votes
1answer
16 views

Find limit inferior and limit superior of $[1+\sin n]$ and $n - [\sqrt n]$

I have to find the limit inferior and limit superior of the following sequences: $$[1+\sin n]$$ and $$n - [\sqrt n].$$ I have done similar exercises before, but never with the integer part function ...
0
votes
1answer
50 views

$\frac{1}{2}\int\frac{1}{1-x^2}d(1-x^2)$

I think the result should be $ln(|1-x^2|)^\frac{1}{2} + C$. But the answer is $ln(1-x^2)^\frac{1}{2} + C$. Could you please tell me where the wrong is? I just used the formula table.
-1
votes
1answer
50 views

Prove serial representation of the integral $\int_0^1 x^x \,{\rm d}x$ [duplicate]

I have to prove the serial representation of: \begin{equation*} \int^1_0 x^x\,{\rm d}x=\sum^{\infty}_{n=0}\frac{(-1)^{n-1}}{n^n}. \end{equation*} It obtains: \begin{equation*} ...
2
votes
3answers
52 views

How do I go about solving the integral of csc x?

So here is a solution provided by another user a couple of years ago and I've seen this solution before but I'm not clear on how or why? What would make me think or tip me off that I should multiply ...
1
vote
0answers
41 views

Primes with the first $k$ digits of the solution of the equation $e^{-x^2}=x$

Let $s$ be the solution of the equation $e^{-x^2}=x$ The first $1000$ digits are : ...
0
votes
2answers
57 views

integration substitution what am I doing wrong?

integrate $\frac{x^3}{(4+x^2)}$ Let $u = 4+x^2$ so $\frac{du}{2}=xdx$ Then I need to integrate $\frac{(u-4)}{u}$ which comes out as $u-4\ln u$ converting back to $x$, $4+x^2-4 \ln(4+x^2)+C$ But I ...
2
votes
3answers
43 views

Prove $\lim_{n\to\infty} \left(\sin \frac{\pi n}{2} \cdot \cos \left(\sin \frac{1}{n} \right ) \right)$ doesn't exist

I must prove that the limit $$\lim_{n\to\infty} \left(\sin \frac{\pi n}{2} \cdot \cos \left(\sin \frac{1}{n} \right ) \right)$$ doesn't exist and also find all of its partial limits. Apparantely ...
3
votes
1answer
30 views

derivative of $f(r\cos\phi,r\sin\phi)=r^a\cos(a\phi)$

Let $$ f(r\cos\phi,r\sin\phi)=r^a\cos(a\phi) $$for some $r\in(0,\sigma)\subset\mathbb R$ and $\phi\in (0,\rho)\subset(0;2\pi]$. How do you calculate $Df=(\partial_1 f,\partial_2 f)$ ? I thought ...
3
votes
4answers
249 views

Finding the limit of a sequence $a_n=\frac{(5n^2)+n+2}{(11n^2)-3}$

What is the simplest way to find the limit of a sequence? I have this sequence $a_n=\frac{(5n^2)+n+2}{(11n^2)-3}$ and I have to find its limit. However I'm lost on how to do this.
6
votes
1answer
135 views

Integration of $\frac{e^{\cos^2x}+\ln(1+x)}{10^{x^3}\arctan(\sqrt{x})}$, possibly numerical

A couple of days ago I came across the following integral: $$\int_{0.02}^{0.08} \frac{e^{\cos^2x}+\ln(1+x)}{10^{x^3}\arctan(\sqrt{x})}\,{\rm d}x$$ The funny thing is, I found this integral written in ...
-2
votes
2answers
75 views

$\int _{ 0 }^{ 1 }{ \frac { { x }^{ t }-1 }{ \ln { x } } dx } $ [duplicate]

How do I solve the following integral: $$\int _{ 0 }^{ 1 }{ \frac { { x }^{ t }-1}{ \ln { x } } dx } $$
12
votes
2answers
520 views

How to avoid stupid mistakes in calculus exams without checking the whole process?

Few days ago I failed my Calculus exams. And again it was mostly due to simple mistakes such as forgetting about minus in front of fraction, switching y coordinates of two points etc. The assignments ...
0
votes
0answers
24 views

Where does the $kΔt$ go?

Frame 142 out of Quick Calculus, 2nd Edition: Suppose the position of an object is given by $S=f(t)=kt^2+lt+S_0$, where $k$, $l$ and $S_0$ are constants. Find $v$. I've worked the problem to: ...
2
votes
4answers
152 views

How to evaluate this indefinite integration $\int \frac{\tan^4 \theta d \theta}{1-\tan^2 \theta}$?

I have to solve this indefinite integration $$\int \frac{\tan^4 \theta d \theta}{1-\tan^2 \theta}$$ I tried it as follows $$I=\int\frac{(\sec^2 \theta-1)\tan^2 \theta d \theta}{1-\tan^2 ...
1
vote
1answer
26 views

Solve the following PDE: $(1+\sqrt{z-y-x})z'_x+z'_y=2$

Solve the following PDE: $(1+\sqrt{z-y-x})z'_x+z'_y=2$ given that $z(x,2x)=2x$. I want to explain to you how we were taught to solve these at class, and this method seemed to work with other ...
-1
votes
1answer
47 views

Hyper 5 lower arithmetic operation. [on hold]

What is the formula for hyper 5 lower? Hyper 4 lower is $x^{x^{y-1}}=z$. Hyper 3 lower is $x^y=z$. Hyper 2 lower is $x*y=z$.
6
votes
1answer
909 views

Laplacians and Dirac delta functions

It is often quoted in physics textbooks for finding the electric potential using Green's function that $$\nabla ^2 \left(\frac{1}{r}\right)=-4\pi\delta^3({\bf r}),$$ or more generally $$\nabla ...
8
votes
5answers
173 views

Why this limit is $-\frac{1}{4}$?

Find $$\lim_{x\rightarrow\infty} \bigg(1-x^{1/2x}\bigg)\cdot\frac{x}{2\ln{x}}$$ I tried this method: $$\begin{align} \bigg(1-x^{1/{2x}}\bigg)\frac{x}{2\ln{x}} & = ...
7
votes
2answers
143 views

Feynman technique of integration for $\int^\infty_0 \exp\left(\frac{-x^2}{y^2}-y^2\right) dx$

I've been learning a technique that Feynman describes in some of his books to integrate. The source can be found here: ...
-2
votes
0answers
28 views

A problem of Taylor series [on hold]

I need a step-by-step solution to the following problem. Sorry, I have nothing done because I don't know how to approach the problem. Determine $\alpha \in \mathbb{R}$ such that $$ \arctan^2 (2x) - ...