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

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0
votes
2answers
31 views

Deduce if the series converges absolutely or conditionally.

$$\sum_{}^\ (-1)^k \frac{(3^k)(k!)^2}{(2k)!}$$ I start by using the absolute convergence test. This eliminates the -1: |1|^k = 1 Then I use the ratio test. $$\left|\frac{3^{k+1} ...
0
votes
0answers
18 views

How to prove whether the series problem converges or diverges?

$Σ$ $(-1)^k$ $ \frac{k^2+3k}{k^3+k+2)}$ I use the absolute value theorem for this problem. Then I use the a limit comparison test on it. $\frac{k^2+3k}{k^3+k+2}*1/k =\frac {k(k^2+3k)}{(k^3+k+2)}→ ...
1
vote
2answers
48 views

Finding the exact amount of a sigma problem?

$$\begin{align*} \sum_{k=1}^\infty \frac{1}{k(k+1)} \end{align*}$$ This is a telescoping series; therefore I use partial fractions to solve. $\int_{1}^{∞} (1)/(k+1) $ = $ ((A/k)+(B)/(k+1))$ A= 1 ...
1
vote
4answers
56 views

without using l'hopital rule

Can someone give me please some guidance hoe to solve the following limit, without using L'Hopital rule? $$\lim\limits_{n \to \infty } \frac{n}{\ln\left(\frac{3n}{5}\right)}$$ Thanks a lot!
0
votes
2answers
24 views

Cauchy sequence in practice

Let $a_n$ be a sequence. Based on Cauchy can I say that if $|a_{n+2}-a_{n+1}| < |a_{n+1}-a_n|$ then $a_n$ converges? The reason behind this is that $a_{n+1}$ is just a small offset of $a_n$ and ...
0
votes
5answers
49 views

Consider the following limit: $\lim_{n \to \infty } \frac{\ln(1+n)-\ln(n^{2})}{\sin(1/n)}$

Can someone give me some guidance on where to begin with the following limit? $$ \lim_{n \to \infty } \frac{\ln(1+n)-\ln(n^{2})}{\sin(1/n)} $$ Thanks!
2
votes
5answers
48 views

Find the center and radius of polar circle equation [on hold]

Find the radius and center of the circle $$r=2\cos \theta+3\sin \theta$$
0
votes
2answers
38 views

Using Taylor series to find $\lim_{x\to 1}\frac{2-(x+3)^{1/2}}{x-1}$ [on hold]

How to find the limit $$\lim_{x\to 1}\frac{2-(x+3)^{1/2}}{x-1}$$ using the Taylor series? I have done the derivatives on function following the regular steps. But all I get is zero.
1
vote
4answers
54 views

Trig differentiation

Prove that there is a constant C such that $$ \arcsin{\frac{1-x}{1+x}} + 2\arctan (\sqrt{x}) = C $$ for all $x$ in a certain domain. What is the largest domain on which this identity is true? What ...
0
votes
0answers
45 views

Integration of a polynomial

I am facing a problem in finding the integral $$\int\frac{r^2}{-C r^3 + r^2 -2 M r +Q^2}\,dr$$ Here M, Q, and C are parameteres (to be fixed later). Could anybody Please help me in finding it? I ...
2
votes
2answers
84 views

Calculation of $\int_0^{\pi} \frac{\sin^2 x}{a^2+b^2-2ab \cos x} dx\;,$

Calculation of $\displaystyle \int_0^{\pi} \frac{\sin^2 x}{a^2+b^2-2ab \cos x} dx\;,$ given that $ a>b>0$ $\bf{My\; Try::}$ Let $\displaystyle I = \int_{0}^{\pi}\frac{\sin^2 ...
0
votes
3answers
72 views

How to solve a differential equation?

I'm trying to solve the system $$\frac{d^4x}{dt}+4x=0,\quad\frac{d^3x}{dt}+x=0.$$ However, I don't know of any method of tackling such a problem. Can anyone please provide a route to a solution? ...
0
votes
1answer
18 views

Sequences with intervals

I'm trying to play a bit with sequences & intervals and I've got a few questions which I'm not sure about: Let $a_n$ be a sequence and $I=(a,b)$ interval such that {$a_n|n\in N$} densed in $I$ ...
0
votes
1answer
16 views

About restricting variables in an integrand, and also changing the look of an integrands.

So, in the last step of, many, integrands, Wolfram chooses to restrict the $x$-values, even if I didn't specify which values $x$ can take on. Take for example: $$\int\frac{dx}{x(x^2-1)^{3/2}} = ...
0
votes
2answers
31 views

infimum and supremum of subsets question

Let $B \subseteq \mathbb{R}_{+}$ such that B is non-empty. consider $B^{-1} = \left \{b^{-1} : b\in B \right \}$. Show that if $B^{-1}$ is unbounded from above, then $\inf\left(B\right)=0$ How can i ...
1
vote
2answers
44 views

Calculating semi axes from given tilted ellipse equation

Hopefully no duplicate of Ellipse $3x^2-x+6xy-3y+5y^2=0$: what are the semi-major and semi-minor axes, displacement of centre, and angle of incline? (see below) Let the following equation $$x^2 - ...
4
votes
1answer
85 views

How to find the maximum and minimum of $\dfrac{\sin x}{x^2+1}$?

How can we find the values of $x$ that give the maximum and minimum of $$\frac{\sin{x}}{x^2+1}$$ I took a lucky guess and found that $\dfrac\pi4$ was fairly close to giving the max, but how does one ...
5
votes
2answers
84 views

Evaluation of $\prod_{n=1}^\infty e\left(\frac{n}{n+1}\right)^{n}\sqrt{\frac{n}{n+1}}$

During my calculation I ended with the following product: $$P=\prod_{n=1}^\infty e\left(\frac{n}{n+1}\right)^{n}\sqrt{\frac{n}{n+1}}$$ I tried to express in term of series by taking the logarithm ...
0
votes
1answer
22 views

how to find a closed form expression for a power series

my question is how do i find a closed form expression for a function f(x) which the power series $\sum_{n=0}^\infty 2n(-7)^n x^{n+2}$ converges to and the value of x for which f(x) equals the given ...
3
votes
3answers
56 views

Is $\int_1^\infty \frac{\log(x-1)}{x(x-1)}\,dx$ convergent?

Does the following integral $$\int_1^\infty \frac{\log(x-1)}{x(x-1)}\,dx$$ converge? If it is convergent can we compute it?
0
votes
1answer
28 views

Counting function for the number of zeros of a continuous positive function?

Let $f(x)$ within $x\in[a,b]$ an absolute continuous function with $f(x)\geq0$ $f(x_m)=0$ for all absolute minima $x_m$ no other zeros than at $x_m$ I am trying to define a counting function for ...
3
votes
3answers
58 views

Problems with this integral $ \int \sqrt{1 + {1 \over t^2} + {2 \over t}} dt$

$$ \int \sqrt{1 + {1 \over t^2} + {2 \over t}}\,\mathrm dt$$ I tried making substitution, using $ u=1 + \dfrac{1}{ t^2} + \dfrac{2 }{ t} $, then , $dt=\dfrac{du}{-2\left({1 \over t^3 }+ {1 \over ...
0
votes
2answers
104 views

Evaluating $\int^{4}_{1} \sqrt{1+\left(\frac{1}{2\sqrt{y}}-7\right)^2} dy$

I was trying to find arc-length of $x = \sqrt{y}-7y$ So basically right now I am stuck with this $$\int^{4}_{1} \sqrt{1+\left(\frac{1}{2\sqrt{y}}-7\right)^2} \,\mathrm dy$$ $$\int^{4}_{1} ...
6
votes
2answers
58 views

Integration $\frac{1}{2\pi}\int_{-\pi}^{\pi}(x-a)^ke^{-i\omega x}dx, \ \ \ \ a\in\mathbb R$.

Give a compact form for the solution of integral: $$\frac{1}{2\pi}\int_{-\pi}^{\pi}(x-a)^ke^{-i\omega x}dx, \ \ \ \ a\in\mathbb R,k\in\mathbb N$$ any suggestions please?
0
votes
2answers
31 views

Write sums in factorial form. [on hold]

Show that in such a way that we can generalize this proof $$(1+2+3+4+5)+(1+2+3+4)+(1+2+3)+(1+2)+1= \binom{7}{3}$$
2
votes
1answer
24 views

Partial Derivative of a nonexistant variable?

I am wondering how I would find the partial derivative of $z = g(r, \theta) = \theta$ with respect to both $r$ and $\theta$. I realize that if you take the partial in respect to $\theta$, it is 1. I'm ...
4
votes
3answers
348 views

Find a value for “c”

For what value of $c$ is $$\lim_{x\to\infty} \left(\frac{x+c}{x-c}\right)^x = e?$$ I am unsure of how to start this question in any sense.
1
vote
0answers
38 views

Antiderivative of $|x − 2| + |x − 3|$ [on hold]

Find the most general antiderivatives of the following function. $$|x − 2| + |x − 3|$$ I started with showing that the antiderivative for $|u|$ is $\dfrac{u|u|}2$. How to proceed then?
0
votes
0answers
18 views

About Calculus Cost, Revenue, and Profit Functions

1.) An international firm produces a new line of product. Their production cost is PhP 250 for each small unit and PhP 350 for each large one. The firm’s production budget for the product is PhP ...
0
votes
2answers
17 views

Find a Cartesian Equation for the Plane Satisfying Those Properties

Find the Cartesian equation of the plan parallel to j and passes through the intersection of the planes described by the equations x + 2y + 3z = 4, and 2x + y + z = 2. I was able to get the ...
0
votes
1answer
135 views

Optimization question on graph

I was having trouble with this question. A triangle has one side parallel to the x-axis, two vertices on the part of the parabola $$y =3 − {x^2\over 12}$$ above the x-axis and the third vertex at ...
2
votes
3answers
64 views

Suppose that $f ' (x)$ exists and $f(x)$ has two roots $x_1$ and $x_2$. Try to prove that:

Suppose that $f'(x)$ exists and $f(x)$ has two roots $x_1$ and $x_2$. Try to prove that: there is $\xi \in (x_1,x_2)$ such that $f(\xi)+f'(\xi)=0$. We cannot use the knowedge of integration.
0
votes
0answers
15 views

Normal derivative property

I have seen in many papers that to obtain some results about PDEs is used the following argument: If $\phi=0$ in $\partial\Omega$ then $\bigtriangledown\phi=\dfrac{\partial\phi}{\partial n}n$, where ...
1
vote
1answer
284 views

Related Rates/ Optimization problem

I was having trouble figuring out this problem. A fisherman is in a boat 3 km from the nearest point A on the coast. The fisherman wishes to return to his camp C located 5 km from the point A. The ...
2
votes
1answer
27 views

Is it true that a quasiconvex, increasing and continous function, is convex?

Let $f:\mathbb R^n \to \mathbb R$ be a continuous and increasing function. Let $f$ be quasiconvex. Let $f(0)=0$. Can we say that $f(x)$ is convex ? If yes, how do we prove it ? Thank you very much ...
1
vote
2answers
148 views

Finding Sum for Infinite Series

Normally when I keep try multiple ways to solve a problem, I get an idea of where to start, and eventually can solve it. But it hasn't been the way for this question and I've been stuck for hours. ...
0
votes
1answer
45 views

proof $ \sum_{x\geq 1}\frac{-\cos(2x)}{x} $ is convergence?

How to proof that $ \sum_{x\geq 1}\frac{-\cos(2x)}{x} $ is convergence? I guess I am suppose to use Dirichlet test but I am really struggling with it.
-3
votes
1answer
51 views

Taking the Derivative of $F(x)=\int_0^x f(t)\,dt$ [on hold]

Let $F(x)=\int_0^x f(t)\,dt$ What is the derivative of $F(x)$? I desperately need guidance!
0
votes
1answer
30 views

Verify Green's Theorem for region bounded by the lines $x=2$, $y=0$, $y=2x$

Verify Green's Theorem for the region D bounded by the lines $x=2$, $y=0$, $y=2x$ and the functions $f(x,y)=(2x^2)y$, $g(x,y)=2x^3$. I have been trying this question for far too long and I can't ...
2
votes
1answer
26 views

Inequality involving Holders Inequalities

Suppose $f\in L^p(\mathbb{R})\cap L^\infty(\mathbb{R})$ for some $p>2$, show that $||f||_{p}\leq ||f||_2^{2/p}||f||_{\infty}^{1-2/p}$ I tried to write $|f|^p=|f|^{\frac{p}{2}}|f|^{\frac{p}{2}}$ ...
0
votes
0answers
11 views

Tips on effectively representing this recurrence relation in a generalized form.

The recurrence relation is $$ y_n = d_1 y_{n-1}+\frac{d}{dx}[y_{n-1}] $$ a good thing to note as well is $$ \frac{d}{dx}[d_n] = d_{n+1} $$ This is terrible to expand out after a good while, The main ...
0
votes
0answers
10 views

What theorems are used in this following proof of derivatives of log normalizer is moments of sufficient statistics?

The below is the derivation of the proof that shows derivative of log normalizer of exponential family is moments of sufficient statistics \begin{equation} ...
1
vote
1answer
65 views

Give an example where $\{a^2_n\}_{n=1}^\infty$ is a Cauchy sequence, but $\{a_n\}_{n=1}^\infty$is not.

Give an example where $\{a^2_n\}_{n=1}^\infty$ is a Cauchy sequence, but $\{a_n\}_{n=1}^\infty$is not. Could somebody give me an example of this? Thanks in advance!
0
votes
0answers
13 views

Accumulated present value of a continuous income stream answer check

Find the accumulated present value of a continuous income stream that earns 4.2% interest annually, when $4000 is deposited per year for 30 years in the account. This is the answer that I got: ...
1
vote
3answers
25 views

Periodic functions and limit at infinity

Why do periodic functions (like $\cos$ or $\sin$ or $\tan$) have no limit at infinity? I can guess that it is because their values don't converge but repeat over and over, but I would like to know ...
1
vote
2answers
27 views

Optimizing a ranch

"A rancher wants to fence in an area of 1000000 square feet in a rectangular field and then divide it in half with a fence down the middle, parallel to one side. What is the shortest length of fence ...
4
votes
6answers
148 views

Show that $\int_0^\infty \frac{\sin (\lambda x)}{e^x} \, \mathrm dx =\frac{\lambda}{1+{\lambda^2}}$

$$\int_0^\infty \frac{\sin (\lambda x)}{e^x} \, \mathrm dx =\frac{\lambda}{1+{\lambda^2}}$$ My intuition telling me there might be an $\arctan$ coming up, but I don't know how to do this ...
2
votes
0answers
79 views

Find $\displaystyle\sum_{k=1}^\infty\dfrac{1}{\left(\binom{2014+k}{2014}\right)}$

Find $\displaystyle\sum_{k=1}^\infty\dfrac{1}{\left(\binom{2014+k}{2014}\right)}$, where $\left(\binom{a}{b}\right)=\dfrac{a!!}{b!!(a-b)!!}$ EDIT : Someone pointed out in the Mathematics chat that my ...
0
votes
1answer
36 views

What is the meaning of “girth” of a rectangular box?

Here's an optimization problem. A parcel delivery service will deliver a package only if the length plus the girth (distance around, taken perpendicular to the length) does not exceed 112 inches. ...
1
vote
1answer
25 views

Proof if $f$ continuous in $x_0$ then there is a neighbourhood of $x_0$ so f bounded.

I have this question : Proof if $f$ continuous in $x_0$ then there is a neighbourhood of $x_0$ so $f$ bounded. I want to know if my proof is valid : If continuous in $x_0$ then : $$\lim_{x \to ...