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

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1answer
58 views

Evaluation of $\prod_{k=1}^{\infty}\frac{a+k^2}{b+k^2}$

While playing around with the question The convergence of a sequence with infinite products, I found Mathematica to give me the result $$ \prod\limits_{k=1}^{\infty}\frac{a+k^2}{b+k^2} = ...
3
votes
1answer
136 views

Prove $\sum\limits_{n=1}^\infty \frac{n!}{3^n\cdot7\times10\times\cdots\times (3n+1)}=\frac{\pi\sqrt3}{2}+\frac32\ln(3)−4$

Prove $$\sum_{n=1}^\infty \frac{n!}{3^n\cdot7\times10\times\cdots\times (3n+1)}=\frac{\pi\sqrt3}{2}+\frac32\ln(3)−4$$
1
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1answer
39 views

Limit Of a Recursive Sequence

Find the limit of the following sequence : $a_1=0$ $a_{n+1}=\frac{a_n^2+5}{4}$ I am trying to get some intuition if the sequence is bounded. $a_2=1.25$ and ...
5
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1answer
98 views

The convergence of a sequence with infinite products

I have a problem to determine convergence (sum over n). $$\sum_{n=0}^\infty \dfrac {a\left( a+1^{p}\right) \ldots \left( a+n^{p}\right) }{b\left( b+1^{p}\right) \ldots \left( b+n^{p}\right) }$$where ...
1
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3answers
66 views

Prove that if $\sum_1^\infty a_n$ converges provided $a_n>0$ for all $n$, then $\sum_1^\infty \sqrt{a_na_{n+1}}$ converges too

I wrote $b_n:=\sqrt{a_na_{n+1}}$ and tried to apply a few tests on the series. Here are the tests that didn't work: $$b_n<=a_n$$ $$\lim_{n\to\infty} \frac{b_n}{a_n}$$ $$\lim_{n\to\infty} ...
0
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1answer
33 views

Why derivative of $(3x^2 - 2)^{\frac{2}{x}}$ can be written as $(3x^2 - 2)^{\frac{2}{x}} \cdot (ln(3x^2 - 2)\cdot\frac{2}{x})'$?

I am struggling with derivatives of exponents functions... Why derivative of $(3x^2 - 2)^{\frac{2}{x}}$ can be written as $(3x^2 - 2)^{\frac{2}{x}} \cdot (ln(3x^2 - 2)\cdot\frac{2}{x})'$? Where does ...
5
votes
5answers
140 views

Is it possible that $(f\circ g)(x)=x$ and $(g\circ f)(x)\ne x$?

Is it possible that $(f\circ g)(x)=x$ and $(g\circ f)(x)\ne x$ In other words, To show $f$ and $g$ are inverse, is it enough to show $(f\text{ o }g)(x)=x$? I have never witnessed a case in which the ...
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0answers
33 views

Prove that for every $n \in \mathbb N$, $\int_{-\pi}^{\pi}f(x)\sin(2nx)dx=0$ if $f(x)$ is odd.

If $f:\mathbb R \to \mathbb R $ is odd continuous function such that $g(x):=f(x + \frac{\pi}{2})$ is even, prove that for every $n \in \mathbb N$, $\int_{-\pi}^{\pi}f(x)\sin(2nx)dx=0$. Since ...
0
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3answers
39 views

Why derivative of $\sqrt[x]{x}$ can be written as $(\exp(\frac{1}{x}\log(x)))'$

I simply cannot understand why the derivative of $\sqrt[x]{x}$ can be written as $(\exp(\frac{1}{x}\log(x)))'$? Also, is that $\log$ the natural log or what?
2
votes
5answers
104 views

How to simplify the integration?

If any integration is in form $$\int \frac{1}{1+x^2}dx$$ it easily follows the $\tan^{-1}x$ but How to simplify if we have $$\int \frac{1}{(1+x^2)^2}dx$$
4
votes
0answers
68 views

Sequences of rapidly decaying analytic functions

Note that we have $$\frac{1}{x}\gg\frac{1}{x^2}\gg\frac{1}{x^3}\gg\cdots\gg e^{-x}.$$ I was wondering if a generalization is possible. Namely, if $f_1\gg f_2\gg f_3\gg\cdots$ are analytic from ...
1
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2answers
56 views

Calculate the sum of $\sum_{n=1}^{+\infty} \frac{1+2^n}{3^n}$

$$\sum_{n=1}^{+\infty} \frac{1+2^n}{3^n} = \sum_{n=1}^{+\infty} \frac{1}{3^n} + \sum_{n=1}^{+\infty} \frac{2^n}{3^n}$$ Each term is geometric series with $-1<r<1$ so they are all covergent. As ...
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0answers
38 views

Surface integral with domain that contains an infinite cone

I'm stuck on the following question: Find $\iint\limits_S {ydS}$ where $S$ is the part of the plane $z=1+y$ that lies inside the cone $z = \sqrt {2({x^2} + {y^2})} $ I tried to parametrize $x,y$ ...
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4answers
81 views

Need assistance solving a limit question without applying l'hopital's rule

This is the question: $$\lim_{x \rightarrow-\infty} \frac{|2x+5|}{2x+5}$$ I know the answer is $-1$, but can someone go through the steps and explaining it to me?
4
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4answers
165 views

Compute $\lim_\limits{n\to\infty}a_n$ where $a_{n+2}=\sqrt{a_n.a_{n+1}}$

I managed to show that the limit exists, but I don't know how to compute it. EDIT: There are initial terms: $a_1=1$ and $a_2=2$.
-2
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1answer
21 views

Calculus linearization have work can't figure out the last part

$$\frac{-7}{86\sqrt{43}}(x-6)+\frac{1}{\sqrt{43}}$$ $$=\frac{-7}{86\sqrt{43}}(x)+\frac{64}{43\sqrt{43}}$$ How did the first answer change to the second answer?
0
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1answer
47 views

Calculus Derivative I have the work for the problem I'm just not sure where one part comes from

I'm new to this so I don't know how the formatting works sorry. So I have all the work for it there is just one thing I don't understand where it is coming from.... I know its a double chain rule ...
0
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3answers
69 views

Calculus limits with sin and cos

$$\lim \limits_{x\rightarrow 0} \frac{\sin(2x)-2x\cos(2x)}{2x-\sin(2x)} $$ I know the answer is 2 I just don't know how to do the work for it.
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4answers
95 views

Integrate $\int\frac{dx}{(x^2+16)^3}$

Solve the following integral: $$\int\frac{dx}{(x^2+16)^3}$$ I have no idea what to do here. I think there will be trigonometric substitution but I can't even seem to get started with this problem. ...
2
votes
2answers
71 views

What is a differential (Calculus)

The differential of a function $y(x)$ is defined as, $$dy = f'(x)dx$$ I didn't know that a differential is actually defined by the above equation and is a function of both $x$ and $dx$, but does ...
1
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1answer
105 views

Expand a function into a Laurent series about a point?

Take the function $f(z)=(z^2+3z+2)e^\frac{1}{z+1}$ We want to expand this into its Laurent series about $z_0$=-1. Alright, so I'm a little confused. This converges everywhere but -1, which throws me ...
2
votes
2answers
70 views

Is it ok to do this change of variable in integration: let $x = x - 1$

In integrals like $\int \sqrt{x-1}\,dx$, is it ok to make this change of variable in integration: "let $x = x - 1$"? It looks sketchy — like saying, let 5 = 4.
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0answers
19 views

Given the formula for magnetic field, if I integrate it with respect to distance, can I plug in values before integrating?

Ok, so lets say I have some function $P(t, r)$ which gives me the magnetic at any radial distance r at time t. If I want to integrate $P(t, r)$ at $t=10s$ with respect to r, can I integrate $P(10, ...
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2answers
54 views

What is the power series for the function $\ln(1+x^3)$? [closed]

How do you find the power series for the function $\ln(1+x^3)$?
0
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2answers
41 views

I have a question about logs and limits in calculus— it should be short

I was looking at $$\int_0^1\left(\frac{1}{\sqrt{s}}\right)^2\ ds.$$ So in calculus, I would evaluate $\ln(1) - \ln(0)$ as the answer. What I don't get and I don't remember why is the answer $\infty$? ...
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1answer
36 views

Evaluating Definite Integrals: $ \int_{-2}^{0} \ (\frac14{t^5}+\frac1 5{t^4}-t) dt$

$$ \int_{-2}^{0} \ (\frac14{t^5}+\frac1 5{t^4}-t) dt$$ My answer is -368/600
3
votes
3answers
72 views

Why the derivative of $n^{1/n}$ is $n^{1/n} \left( \frac{1}{n^2} - \frac{\log(n)}{n^2}\right)$

Why the derivative of $n^{1/n} = \sqrt[n]{n}$ is $n^{1/n} \left( \frac{1}{n^2} - \frac{\log(n)}{n^2}\right)$ (according to Maxima and other tools online)? I have tried to applied the chain rule, but ...
0
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1answer
31 views

The Center of Mass of a solid quadrant.

I have found via integration that the y coordinate is $$y =h/2 = 120 mm$$. The x coordinate is $$x = \frac{-4r}{3\pi} = -51.9mm$$ and the z coordinate is $$z = r - \frac{4r}{3\pi} = 69.1 mm$$. I have ...
0
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0answers
28 views

Find the supremum of the function

Hi I'm trying to figure out for which values of $w$ the absolute value of the supremum of $u(x,t)$ is infinity. The function $u(x,t)$ is the following. According to my calculation is going to be all ...
0
votes
1answer
30 views

What's the next step here to understand the existence and uniqueness theorem?

I'm trying to find an intuitive grasp of the existence and uniqueness theorem, but all the explanations online explain it using higher mathematics, which sort of defeats the purpose of explaining... ...
3
votes
3answers
64 views

How can I integrate $\int {dx \over \sqrt{3^2+x^2}} $ using Trigonometric Substitution?

$$\int {dx \over \sqrt{9+x^2}} = \int {dx \over \sqrt{3^2+x^2}} $$ $$ x =3\tan\theta$$ $$dx = 3\sec^2\theta$$ $$\int {3\sec^2\theta \over \sqrt{3^2 + 3^2\tan^2\theta}} d\theta$$ $$\int {3\sec^2\theta ...
1
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2answers
75 views

Orthogonality lemma sine and cosine

I want to know how much is the integral $\int_{0}^{L}\sin(nx)\cos(mx)dx$ when $m=n$ and in the case when $m\neq n$. I know the orthogonality lemma for the other cases, but not for this one.
0
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0answers
45 views

find the supremum

Hi I'm trying to figure out for which values of $w$ of $u(x,t)$ the absolute value of the supremum of $u(x,t)$ is infinity. The function $u(x,t)$ is the following. According to my calculation is ...
0
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2answers
180 views

Center of mass of the semi cylindrical shell.

I have to find the x and z co-ordinates of the centre of mass. Here is finding the x; $$V \bar{x} = \int_{-l}^0x dV$$ $$\implies \pi r \delta r l \bar{x} = \int_{-l}^0x\pi r \delta r \delta x$$ ...
0
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0answers
41 views

Derive the formula to rotate a bounded region about y = x.

Derive the formula to rotate the region bounded by: $$y = x + \frac1x \text{ and } y = 4$$ about $y = x$. I understand that you do not simply subtract $x$ from both functions and use the washer ...
2
votes
1answer
206 views

Surface integral over parabolic cylinder that lies inside another cylinder

To be precise, I'm given the following: Find $\iint_K {xdS}$ over the part of parabolic cylinder $z = \frac{{{x^2}}}{2}$ that lies inside the first octant part of the cylinder $x^2+y^2=1$. In ...
2
votes
3answers
69 views

Does $\int_1^2 \frac{\ln(x)}{x-1} dx$ converge and what test is used?

$$\int_1^2 \frac{\ln(x)}{x-1} dx$$ How does one determine convergence of this? I am not interested in the value of it. I tried comparing to $1/(x-1)$ but the integral related to that diverges, and I ...
1
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2answers
93 views

On a connection between Newton's binomial theorem and general Leibniz rule using a new method.

In calculus the general Leibniz rule asserts that Let $n$ be a natural numbers, if $f$ and $g$ are $n$-times differentiable functions at a point $x$, then the function $fg$ is also $n$-times ...
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1answer
96 views

How to integrate $\int \frac{1}{\cos(a) - \cos(x)}\mathrm{d} x$ [closed]

How do I integrate $$\int \frac{1}{\cos(a) - \cos(x)} \mathrm{d} x?$$ Is there a a substitution or any other method? Please help
2
votes
3answers
88 views

Inner product: $(x,z)=(y,z)\implies x=y$?

We've talked about inner products in our last tutorial and couldn't really get answered the following questions: Let $(\cdot,\cdot)$ be any inner product. If $(x,z)=(y,z)$ for all $z$ of any given ...
0
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1answer
51 views

Finding the definite integral using two variables - what am I doing wrong here?

I'm trying to find the average value of the function: $$p(t) = t7*sin0.2t^2+75 \quad dt \quad on[0,12]$$ So I wanted to start off by first finding the definite integral. I'm being thrown off by the ...
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2answers
117 views

Find: $\int_0^{\infty}\frac{\sinh x}{1+\cosh^2x}dx$

$$\int_0^{\infty}\frac{\sinh x}{1+\cosh^2x}dx$$ Here's what I've attempted: Using the identity $1+\cosh^2x=\sinh^2x$ I got: $$\int_0^{\infty}\frac{\sinh x}{\sinh^2x}dx=\int_0^{\infty}\frac1{\sinh ...
1
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1answer
36 views

Limit of a sequence = subsequence of a limit subsequence

Let there be a sequence $a_n$ and for all subsequence of it $b_n$ there is a subsequence $c_n$ that convergence to $a$ Prove: $a_n \rightarrow a$ Where should I start?
1
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1answer
29 views

What are the correct limits of integration here?

Say we have a differential equation in the following form and we want to integrate both sides to solve for P. $$\frac{1}{P}dp = f(x)dx$$ The domain of $x$ is $(-\infty, \infty)$. The domain of $P$ ...
1
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0answers
19 views

Implicit Equation: Solve for boundary of integral

Let $G(J)$ denote the CDF of $J$. I have an equation of the form $$ G(V-k) A = k + \int_a^{V-k} J dG(J)$$ for some constants $A$, $k$, $a$. Is there a way I could get an explicit solution for $V$ ...
10
votes
1answer
223 views

Exercise about continuous functions

Consider a continuous function $f \, : \, [0,1] \, \longrightarrow \, [0,+\infty)$ such that $f(0)=f(1)=0$ and : $\forall x \in (0,1), \; f(x) > 0$. I would like to prove that there exist ...
1
vote
1answer
80 views

Evaluate a double integral bounded by two circles

Evaluate the integral $\iint_R y \ dR$ where $D$ is a region between the circles $x^2+y^2=2x$ and $x^2+y^2=4$ and on the first quadrant. Is my answer true? ...
5
votes
1answer
60 views

Find the value of $\lim_{n \rightarrow \infty} \Big( 1-\frac{1}{\sqrt 2} \Big) \cdots \Big(1-\frac{1}{\sqrt {n+1}} \Big)$

$$\lim_{n \rightarrow \infty} \Big( 1-\dfrac{1}{\sqrt 2} \Big) \cdots \Big(1-\dfrac{1}{\sqrt {n+1}} \Big)$$ Attempt: Let $y = \lim_{n \rightarrow \infty} \Big( 1-\dfrac{1}{\sqrt 2} \Big) \cdots ...
4
votes
5answers
188 views

Convergence of the integral $\int_0^{\pi/2}\ln(\cos(x))dx$

I want to Show that whether the integral $$\int\limits_0^{\pi/2}\ln(\cos(x))dx$$ is convergent ot not. My Approach: Let $y=cos(x)$, then the above integral reduces to ...
1
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0answers
31 views

Continuity of operator.

For a fixed given function $f:[0,1]\times \mathbb{R} \rightarrow \mathbb{R}$ which is assumed to be measurable and bounded (i.e. $f\in L^\infty([0,T]\times \mathbb{R})$). Let $U\subset \mathbb{R}$ be ...