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

learn more… | top users | synonyms

4
votes
5answers
74 views

Calculate $\lim_{n \to \infty} \ln \frac{n!^{\frac{1}{n}}}{n}$

How can I calculate the following limit? I was thinking of applying Cesaro's theorem, but I'm getting nowhere. What should I do? $$\lim_{n \to \infty} \ln \frac{n!^{\frac{1}{n}}}{n}$$
1
vote
1answer
12 views

About Fourier transform and complex conjugate

why this passage is correct ? \begin{equation*} \mathscr{F}[h(-\tau)] = H^*(f), \end{equation*} when $h(\tau)$ is a real function of real variable $\tau$, and $H^*(f)$ is the complex conjugate of ...
0
votes
1answer
20 views

Check if the following are perpendicular.

I have these expressions : $$2x+2y-5=0 \\ x=3-t,y=2+t,z=1-3t$$ I need to check if they are perpendicular. This is what I did : The following vectors represent the expressions $\langle ...
1
vote
3answers
33 views

Prove that $f : [a,b] \rightarrow \mathbb{R}$ is a bijection from $[a, b]$ to $[f(a), f(b)]$

I'm a 1st year mathematics student, and in my analysis class I'm having trouble with proving the following: Let $a < b \in \mathbb{R}$, and let $f : [a,b] \rightarrow \mathbb{R}$ be a continuous ...
2
votes
1answer
40 views

For which values of $x$ is the following series convergent: $\sum_0^\infty \frac{1}{n^x}\arctan\Bigl(\bigl(\frac{x-4}{x-1}\bigr)^n\Bigr)$

For which values of $x$ is the following series convergent? $$\sum_{n=1}^{\infty} \frac{1}{n^x}\arctan\Biggl(\biggl(\frac{x-4}{x-1}\biggr)^n\Biggr)$$
1
vote
1answer
77 views

Extreme values of a continuous function on a closed connected domain

Suppose a one-variable continuous function has only one extreme value on a closed interval and it is a local minimum, we can prove it is the global minimum on the interval. Suppose a one-variable ...
4
votes
2answers
500 views

Proof that Newton Raphson method has quadratic convergence

I've googled this and I've seen different types of proofs but they all use notations that I don't understand. But first of all, I need to understand what quadratic convergence means, I read that it ...
2
votes
1answer
16 views

Evaluating a triple integral by insepction

I would like to evaluate the triple integral: $$\iiint\limits_D {2 + 3{x^2} + 3{y^2}dV}$$ where $D$ is a conic domain with vertex $(0,0,b)$ and axis along the $z$-axis with a base (disk) with radius ...
5
votes
2answers
93 views

Infinite integrals$\int_0^{ + \infty } {\frac{1}{{\left( {x + 1} \right)\left( {{x^n} + 1} \right)}}dx} .$

How to calculate $$\int_0^{ + \infty } {\frac{1}{{\left( {x + 1} \right)\left( {{x^n} + 1} \right)}}dx} .$$
2
votes
1answer
27 views

Are the extrema of this function global or local?

Last question about this function, I promise. The function $f: \mathbb R \rightarrow \mathbb R$ is given by $$f(x) = \begin{cases} \frac{x^2+5x+7}{x+3} & \mathrm{for} \; x < -3 \\ 0 & ...
1
vote
1answer
73 views

Feynman Integration Problem

$$ I = \frac{\pi^2}{8} - \int_0^1 \frac{\arctan(x)}{\sqrt{1-x^2}} \,dx $$ Evaluate $I$ $$ I = \frac{\pi^2}{8} - \int_0^1 \frac{\arctan(x)}{\sqrt{1-x^2}} \,dx$$ $$f(a) = \int_0^1 ...
-2
votes
1answer
38 views

Integration by parts prove integral of cos^n x dx [on hold]

I'm having a problem with one of my questions. How can I prove that $\begin{align}\int\cos^n x dx&=\sin x\cdot\cos^{n-1}x+(n-1)\int\sin^2x\cos^{n-2}x dx\end{align}$ ?
1
vote
1answer
35 views

Let $a_n>0$ for $n \geq 1$ and let series: $\sum_{n=1}^{\infty}a_n$ diverge. Let $S_n=a_1+a_2+…+a_n > 1$ for $n \geq 1$

Prove that the series: $$\sum_{n=1}^{\infty}\frac{a_{n+1}}{S_n \ln S_n}$$ diverges and the series : $$\sum_{n=1}^{\infty}\frac{a_{n}}{S_n \ln^2 S_n}$$ converges. (Using the famous criteria I ...
1
vote
1answer
21 views

For what $\alpha$ does the integral absolutely and for what conditionally converge?

For what $\alpha$ does the integral absolutely and for what conditionally converge ? $$\int_{0}^{1}\frac{\ln^{\alpha} (1+x^4)}{x^4}\cos{1 \over x}dx$$ Not sure which criteria to use to prove the ...
0
votes
0answers
13 views

How to express $[u,v,w]$ as a function of $\phi,\theta$ and the norms of the vectors?

$u,v$ are linearly independent and $w$ is a non-zero vector. Let $Angle(u,v)=\phi$ and $Angle(u \times v,w)=\theta$. Express $[u,v,w]$ as a function of $\phi,\theta$ and the norms of the vectors. ...
0
votes
1answer
27 views

Fourier Transform of sin function

Hi there I'm trying to find the fourier transform of the following: \begin{equation*} x(t) = \sin(\pi t + \pi). \end{equation*} From what I know, I would integrate this using: $FT =\int x(t)e^{-iwt} ...
2
votes
2answers
27 views

How to find the length between 2 points given a pivot

I am not great at math but I have done the previous steps to my problem. This is the last step where I need to find out the distance between C,D. I am writing a program that will output this ...
-4
votes
1answer
52 views

Extreme values of a function … [on hold]

Suppose a one-variable continuous function has only one extreme value on a closed interval and it is a local minimum, we can prove it is the global minimum on the interval. Suppose a one-variable ...
-2
votes
1answer
25 views

Is integration an injective operation on the set of (integrable) functions? [on hold]

Is indefinite integration an injective operation on the set of (integrable) functions? Thanks Alex
-3
votes
0answers
16 views

find the inverse Laplace transform of complex function3 [on hold]

It would be appreciate if someone help me to obtain the inverse Laplace transformation of the complex function F(s) is $$ 1\over\ \sqrt{s^{2}+1} $$ Thanks.
2
votes
1answer
75 views

Calculate $\lim_{n \to \infty}(\sin nx)^\frac{1}{n}$

I know that $$(n)^\frac{1}{n} \to 1$$ and $(a)^\frac{1}{n} \to 1$ (with $a \in \mathbb{R+}$). However, I was wondering what can be said about $$\lim_{n \to \infty }(\sin nx)^\frac{1}{n}$$ and, more ...
4
votes
0answers
33 views

A tough limit problem involving $1/(\sin x - \sin a)$ and its generalization

Long back I had encountered the following problem in Hardy's Pure Mathematics (originally from the infamous Mathematical Tripos 1896): If $$f(x) = \frac{1}{\sin x - \sin a} - \frac{1}{(x - a)\cos ...
3
votes
6answers
205 views
+200

The box has minimum surface area

Show that a rectangular prism (box) of given volume has minimum surface area if the box is a cube. Could you give me some hints what we are supposed to do?? $$$$ EDIT: Having found that for ...
5
votes
3answers
91 views

Difficult Coordinate Geometry and Calculus Question

I was given this question by a friend and after tirelessly working on it I have not come up with anything substantial. I was hoping someone in the community could provide a pointer or possibly a ...
-5
votes
3answers
38 views

Mean value theorem Problem? [on hold]

Using the "Mean value theorem" prove that $\tan(x)>x$ for $0 < x < \frac{\pi}{2}$
3
votes
0answers
46 views

Integral formulas involving continued fractions

Ramanujan posed the following formulas as questions in the Journal of Indian Mathematical Society: $$\int_{0}^{\infty}\dfrac{\sin nx\,\,dx}{{\displaystyle x + \dfrac{1}{x +}\dfrac{2}{x +}\dfrac{3}{x ...
76
votes
8answers
4k views

Evaluating $\lim_{n\to\infty} e^{-n} \sum\limits_{k=0}^{n} \frac{n^k}{k!}$

I'm supposed to calculate: $$\lim_{n\to\infty} e^{-n} \sum_{k=0}^{n} \frac{n^k}{k!}$$ By using W|A, i may guess that the limit is $\frac{1}{2}$ that is a pretty interesting and nice result. I ...
0
votes
2answers
43 views

Expressing volume of the wine in the tank as a function of height of surface from bottom of the tank? [closed]

I'm stuck on this question. Suppose the wood nymphs and satyrs are having a hot party in honor of Bacchus and the wine is flowing freely from the bottom of a giant cone-shaped barrel which is $12$ ...
3
votes
3answers
111 views

Determine if a series defined by cases is convergent and calculate the sum

Consider $\sum_{n=1}^\infty a_n$, where $a_n$ is $$3^{-n}$$ if $n$ is even and $$\ln \frac{(n+2)(n+1)}{n(n+3)}$$ if $n$ is odd. I have to say if it is convergent and calculate its sum, but the ...
1
vote
1answer
19 views

Volume inside region delimited by surfaces $z=5-x^2$, $z=y$ and $y=1$.

I need to find the volume inside the region $E$ delimited by surfaces $z=5-x^2$, $z=y$ and $y=1$. I've spent few hours on this and would really need a hint from a charitable soul. I see that the ...
7
votes
7answers
536 views

What mistake have I made when trying to evaluate this limit?

Suppose $a$ and $b$ are positive constants. $$\lim \limits _ {n \to \infty}n - \sqrt{n+a} \sqrt{n+b} = ?$$ What I did first: I rearregended $\sqrt{n+a} \sqrt{n+b} = n \sqrt{1+ \frac{a}{n}} \sqrt{1+ ...
4
votes
3answers
63 views

How do I solve the following differential equation

$$\frac{d^2y}{dx^2}=x^2y$$ Solving it by writing out a characteristic equation is not helping me find the solution to the above equation. Any help would be appreciated thanks.
1
vote
1answer
458 views

Cauchy's definition of limit and Heine's definition of limit are equivalent..

Proof: Cauchy -> Heine $$ \forall x\epsilon D\ whiteout \ a\ |x-a|<\delta\Rightarrow |f(x)-L|<\varepsilon \\ \exists n_0\epsilon\mathbb{N} \ \forall n\ge n_0 \ |a_n-a|<\delta \Rightarrow ...
1
vote
1answer
29 views

Parameterization of the curve of intersection of a surface

I need to parameterize the curve of intersection of the surface: $x^2+y+z=2$ and $xy+z=1$ What I've done so far is said: $z=2-x^{2}-y$ therefore $xy+(2-x^{2}-y)=1$ (using substitution). Then, we ...
2
votes
6answers
95 views

Calculate the sum of three series which may be telescoping

Let $$\sum_{n=1}^\infty \frac{n-2}{n!}$$ $$\sum_{n=1}^\infty \frac{n+1}{n!}$$ $$\sum_{n=1}^\infty \frac{\sqrt{n+1} -\sqrt n}{\sqrt{n+n^2}}$$ I have to calculate their sums. So I guess they are ...
2
votes
1answer
24 views

Finding volume using washer method

I'm supposed to determine the volume of the region obtained by revolving the region lying below the graph of the given function and above the $x$-axis about the specified axis. The problem I'm given ...
0
votes
2answers
68 views

Name of a limit theorem [on hold]

What is the name of the theorem below? I have tried googling it with no luck.
-2
votes
0answers
12 views

Applications of the Define Integral - Volume by moving square

The point of intersection of the diagonals of a square is in motion along the diameter of a circle of radius a; the plane in which the square lies remains perpendicular to the plane of the circle, ...
-3
votes
1answer
35 views

Does $\sum a_n$ converge if $a_1 = 1$ and $a_{n+1}=\frac{2+\cos n}{\sqrt n}a_n$ [on hold]

Let $$a_1 = 1$$ and $$a_{n+1}=\frac{2+\cos n}{\sqrt n}a_n.$$ Consider $$\sum a_n.$$ How do I calculate if the series converges? The definition by recurrence troubles me a lot.
1
vote
3answers
37 views

Integral Question using the Rule of Subsitution

I'm confused as to why $ \int e^{kx}dx$ = $\frac{e^{kx}}{k} + C$. I'm using the rule of substitution and came to the conclusion that it should be $e^{kx}k$ because the derivative of $kx$ is $k$. What ...
1
vote
1answer
22 views

Find real numbers that makes two different equations the same curve

I'm taking a vector calculus course, and I'm having trouble with this question. I know that I can make $B$ into $A$ (for the $x$ component) by multiplying $X$ by $t^3$ , which will be $t + t^3$, but ...
0
votes
2answers
39 views

I need help with the integration order please

the integral is as follows: find the volume between these regions bounded by : $z = x^2 + 3y^2$ and $z = 9 - x^2$ I discovered that this would be the space bounded by the elliptic paraboloid and the ...
1
vote
3answers
49 views

Derivative of a Rational function $f(x)=\sqrt{2x-5\over3x+1}$

I'm trying to find the derivative of, $$f(x)=\sqrt{2x-5\over3x+1}$$ I think I can change this into $$f(x)= \left({2x-5 \over 3x+1}\right)^{1\over2} \\ =[(2x-5)(3x+1)^{-1}]^{1 \over 2}$$ Am I not ...
0
votes
0answers
21 views

Skellam CDF Increasing vs Decreasing in a parameter

I'm working with the following Poisson difference distribution: $$\text{Prob}\{X_1-X_2 \geq 0\} $$ where $X_1 \sim$ Poisson $(\mu_1)$ is independent from $X_2 \sim$ Poisson $(\mu_2)$. I need to ...
0
votes
1answer
72 views

Geometric series for this problem..?

I am trying to work this out with geometric series: ...
1
vote
2answers
28 views

Is this an open set in $\mathbb R^2$?

Is $\{(x,y)\mid y = \sin \frac {1}{x}, x>0\}$ an open set? (It is living in $\mathbb R^2$.) I think it should be open because $(0,0)$ seems to be a limit point of this set while it is not an ...
0
votes
0answers
7 views

convergence improvement

I try working in chevyshef series of zeta[2] but althougth the convergence it is not to bad it is hard to compute it would posible to split the sum? $$\zeta (2)=\sum _{n=1}^{\infty } \frac{(-1)^{2 n} ...
1
vote
2answers
60 views

Proof that a degree $4$ polynomial has a minimum

Let $$f(x) = x^4+a_3x^3+a_2x^2+a_1x+a_0.$$ Prove that $f(x)$ has a minimum point in $\Bbb{R}$. The Extreme Value Theorem implies that the minimum exists in some $[a,b]$, but how do I find the ...
1
vote
1answer
25 views

Deciding whether $(ax+b)/(cx+d)$ is increasing

In order of studying usual function such us : 1) $f(x)=ax^2$ 2) $g(x)=\frac{a}{x}$ 3) $h(x)=\frac{ax+b}{cx+d}$ 4) $p(x)=ax^2+bx+c$ To know if a function is increasing or decreasing we can ...
2
votes
1answer
31 views

Find all asymptotes of the function

Follow-up on Is the function continuous and differentiable at $x=-2$? The function $f: \mathbb R \rightarrow \mathbb R$ is given by $$f(x) = \begin{cases} \frac{x^2+5x+7}{x+3} & \mathrm{for} \; ...