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

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3
votes
2answers
59 views

How can I find the convergence/divergence of $\sum_{n=1}^\infty {n! \over n^n}$

Using either root test or ratio test. I have the feeling that it is the root test, I'm not sure how to proceed from this: $$ \sqrt[n]{n! \over n^n}= {(n!)^{1\over n} \over n} $$
1
vote
1answer
53 views

Evaluate $\int_{0}^{1} \frac {\ln x}{1-x^2} \mathrm{d}x $

I found this question in a reference book: $$\int_{0}^{1} \frac {\ln x}{1-x^2} \mathrm{d}x $$ Can Anyone give me Idea how do I begin solving this?
0
votes
3answers
35 views

Taylor Series confusion [on hold]

Can someone explain how to do this problem? I just started Taylor series and I am super confused on how to do it.
4
votes
1answer
42 views

Is this differential equation separable?

$$x\frac{dy}{dx}-y^2 = \frac{dy}{dx}+5$$ I have found that this equation is differentiable as shown in the following. $$x\frac{dy}{dx}-\frac{dy}{dx} = y^2+5$$ $$dy(\frac{x}{dx}-\frac{1}{dx}) = ...
1
vote
3answers
67 views

How can I find if $\sum_{n=1}^\infty {n! \over 10^n} $ converges or diverges?

$$\sum_{n=1}^\infty {n! \over 10^n} $$ I wasn't sure on which method to use, I think the ratio test might work, but I'm stuck. Here's what I have so far: $a_n$= $n! \over 10^n$ & ...
0
votes
0answers
18 views

Calculus 3 Spherical coordinates: I'm not sure how to set this up.

find the volume of the region enclosed by the sphere x^2+y^2+z^2=324 and the cylinder (x-9)^2+y^2=81 by using spherical coordinates. I'm just not seeing how to convert this into a form where spherical ...
1
vote
1answer
37 views

Proof that $\lim \frac{a_n}{1+a_n^2} = 0 \implies \lim a_n = 0$

I´ve tried some exercises about sequences convergence, particularly: Let $a_{n}$ be a sequence such as $\displaystyle\lim_{n\rightarrow\infty}\frac{a_{n}}{1+a_{n}^2}=0.$ Prove that $a_{n}$ ...
0
votes
1answer
20 views

tangents parametric equation

$\left\{ \begin{array}{rl} x(t)=t^2 +1 \\ y(t)= t^3 -1 \end{array} \right.$ i) show for every t_0 not 0 that the given curve got a tangent line at $(x(t_0)),y(t_0))$ and find a parametric ...
0
votes
1answer
15 views

Interval of convergence for a series

I am currently trying to determine the interval of convergence, but I keep getting 0 for all my questions. I have attached one of the questions that I am unable to solve completely and I would really ...
0
votes
1answer
45 views

Does this compound interest problem coincide to the value of e by coincidence?

An account starts with €$1.00$ and pays $100\%$ interest per year. If the interest is credited once, at the end of the year, the value of the account at year-end will be €$2.00$ . What happens if the ...
2
votes
5answers
93 views

How we can solve that $\lim _{_{x\rightarrow \infty }}\int _0^x\:e^{t^2}dt$?

How we can solve that $\lim _{_{x\rightarrow \infty }}\int _0^x\:e^{t^2}dt$ ? P.S: This is my method as I thought: $\int _0^x\:\:e^{t^2}dt>\int _1^x\:e^tdt=e^x-e$ which is divergent, so all your ...
2
votes
2answers
47 views

How to find convergence/divergence of this series

$$\sum_{n=1}^\infty {1+\cos(n) \over n^2}$$ I used the comparison test and said that $\sum_{n-1}^\infty {1 \over n^2}$ is comparable and also larger than $\sum_{n=1}^\infty {1+\cos(n) \over n^2}$, ...
2
votes
1answer
74 views

Showing that $\sum\limits_{n=1}^{\infty} (a_1+2a_2+…+na_n)/n(n+1) = \sum\limits_{i=1}^n a_n $

Let $\sum\limits_{n=1}^{\infty} a_n$ a series of positive terms convergent. Show that $\sum\limits_{n=1}^{\infty} \frac{a_1+2a_2+...+na_n}{n(n+1)}$ converges to the same value of $\sum\limits_{i=1}^n ...
4
votes
4answers
125 views

How to prove that the value of $e$ is irrational without using the number $e$ itself [on hold]

Recently I have tried to prove that the value of $e$ is irrational without using the number $e$ itself. I have seen that the number $e$ can be expressed as $$\lim_{n\to\infty}(1 + 1/n)^n;$$ however, ...
1
vote
0answers
19 views

Bound of integration over the surface area?

Compute the surface area of that portion of the sphere $x^2+y^2+z^2=a^2$ lying within the cylinder $x^2+y^2=ay$ where $a>0$ I first parametize the sphere using spherical coordinate. I think ...
1
vote
0answers
26 views

How would I determine if this infinite series is convergent or divergent using the limit comparison test?

$$\sum_{n=1}^\infty {2^n \over3+4^n}$$ My thinking is that $4^n$ will grow much more rapidly than $2^n$, and the +3 in the denominator is negligable. Therfore, I should compare it to ...
5
votes
3answers
93 views

$\lim_{n \to \infty} \frac{1}{2n}\log\binom{2n}{n}$ [duplicate]

Please help me to solve this problem. I can find almost no clue regarding the log part. I tried to break the $\binom{2n}{n}$ part, but in vague...will the breaking help me in anyway?
0
votes
2answers
25 views

In differentiability

Let $f(x,y)$ and $g(x,y)$ are differentiable functions in $x$ and $y$. Suppose $f(x,y) = F(g(x,y))$.My question, Is $F$ differentiable function?!.
-5
votes
0answers
30 views

Maclaurin series C++ [on hold]

Im really new to C++ and this is my problem. Please help:( Thanks a lot P/s: an error of 0.01, the picture was cut off
3
votes
0answers
54 views

What is the “$\cdot$” in this: $\left\langle \nabla f,\:\cdot\right\rangle$

I've read the Wikipedia page on exterior derivatives, and it states the following: \begin{align} df&=\sum_{i=1}^n\frac{\partial f}{\partial x_i}dx_i=\left\langle \nabla ...
1
vote
1answer
22 views

Limit of a set of fractions

I'm having trouble with this particular exercise in limits, and I just can't seem to find a way to crack it. I saw a similar exercise online where they used integrals, but it's pretty early in the ...
2
votes
1answer
53 views

How we can find the sign for trigonometric functions without graph

For $\sin(x)$ or $\cos(x)$ etc. how we can show that it is negative on $\left[\pi ,2\pi \right]$ ? without graph? So if we have $\sin(2x)$ or $\cos(2x)$ how we can find the sign on $\left[0,2\pi ...
0
votes
3answers
61 views

Integrate problem

We have to integrate $\int _0^{\pi }\:\left|\sin\left(2x\right)\right|dx$ and in my book, they split integral: $\int _0^{\pi }\:\left|\sin\left(2x\right)\right|dx=\int _0^{\frac{\pi ...
4
votes
1answer
65 views

Let a be a positive number. Then $\lim_{n \to \infty}[\frac{1}{a+n}+\frac{1}{2a+n}+\cdots +\frac{1}{na+n}]$

Problem : Let $a$ be a positive number. Then $$\lim_{n \to \infty}\left[\frac{1}{a+n}+\frac{1}{2a+n}+\cdots +\frac{1}{na+n}\right]$$ Please suggest how to proceed in such limit problems, will be of ...
0
votes
1answer
12 views

Solving for inverse transformation in change of variables

This is something probably silly, but I don't seem to see how this works. I have 2 independent random variables $X \sim Gamma(\alpha, \beta_x)$ and $Y\sim Gamma(\alpha, \beta_y)$, and I need to show ...
1
vote
2answers
138 views

Is this integral impossible to solve?

Is possible to express the antiderivative $$\int\frac{-3e^{-x^3}}{x^2}dx$$ in terms of elementary functions?
-1
votes
1answer
47 views

Compute $f_x(0,0)$ etc. for the following function $f(x,y))$

Let $$f(x,y) = \begin{cases} xy \frac{x^2 - y^2}{x^2 + y^2} \text{ when } (x,y) \neq (0,0) \\[2ex] 0 \text{ when } (x,y)=(0,0) \end{cases}$$ Compute $f_x(0,0)$, $f_y(0,0)$, $f_{xx}(0,0)$, ...
-2
votes
0answers
27 views

Programming language for function analysis

what would you suggest as a programming language for working with functions and graphs. I need them to work with complex piecewise functions. P.S it is okay, if you suggest any software.
1
vote
1answer
29 views

Multiplication and division under integral sign for non-constant function: $\int f(t)dt \iff \int f(t)\frac{g(t)}{g(t)}dt$?

Suppose that you have to evaluate an integral $$\int f(t)dt.$$ Is it allowed to multiply and divide under the integral sign for the same non-constant function? That is, is it allowed to write: ...
0
votes
2answers
28 views

Finding the Jacobian of non square inverse matrix

I am trying to use the Newton method to a non square system. The jacobian matrix is not square and I cannot inverse it. Can anyone help? How can I calculate this 'inverse'?.
0
votes
2answers
39 views

Two-sided limits using advanced algebra

Why does $\frac{1-x}{x-1}=-1$? This is part of a limits problem I am solving using algebra. Thanks.
0
votes
1answer
38 views

How do you compute the length of a curve $y^2=x^3 + x^2$

So I came across this question: consider the closed curve C defined by $y^2 = x^3 + x^2\ (-1\le x\le 0)$ i) compute the length of the curve ii) find the area of the surface formed by revolving the ...
0
votes
4answers
40 views

Eliminating $t$ in the solution of a Differential Equation

My task is to show that the trajectories of the system: $\frac{dx}{dt}=y$, $\frac{dy}{dt}=x$ are hyperbolas given by $H(x,y)=y^2-x^2=c.$ Solving the above system I got: $x=c_1e^t+c_2e^{-t}$, ...
1
vote
4answers
35 views

Proving that $\lim_{n\rightarrow\infty}n(a_{n+1}-a_{n})=1 \implies a_n$ diverges to $\infty$

I'm trying to prove that given a sequence $a_{n}$ such as $\displaystyle\lim_{n\rightarrow\infty}n(a_{n+1}-a_{n})=1,$ then $a_{n}$ diverges to $\infty.$ I'm lost searching a path to prove it. I ...
1
vote
1answer
38 views

Sum of absolute values is finite

Suppose $\lambda_m \in \mathbb{R}$ and suppose that $\sum_{m \in \mathbb{N}} \lvert i + \lambda_m \rvert^{-p} < \infty$. Then why does $$\sum_{(m,n) \in\mathbb{N}\times\mathbb{Z}} \left\lvert i \pm ...
-1
votes
1answer
77 views

Integral identity involving sin(x)/x

Prove or disprove $$\displaystyle\int_{-\infty}^{\infty} \frac{3 \sin \left( x\right )}{x} \mathrm{d}x = \int_{-\infty}^{\infty} \frac{4 \sin ^ 3\left( x\right )}{x^3} \mathrm{d}x$$
3
votes
1answer
47 views

How we can show that $\:I_n\ge \frac{2}{\pi }\left(\frac{1}{n+1}+\frac{1}{n+2}+…+\frac{1}{2n}\right)$

We have $I_n=\int _{\pi }^{2\pi }\:\frac{\left|sin\left(nx\right)\right|}{x}\:dx,$ and we need to show that$\:I_n\ge \frac{2}{\pi }\left(\frac{1}{n+1}+\frac{1}{n+2}+...+\frac{1}{2n}\right)$ I write ...
0
votes
0answers
34 views

Question about convergence in $L^2$ (revisited)

Yesterday I asked the folowing question: Question about convergence in $L^2$ which was answered negatively with a counterexample. Here, I wonder if one can find the right set to look at: Assume we ...
10
votes
3answers
116 views

Proving the limit of a nested sequence

I have trouble proving the next sequence limit: $\displaystyle\lim_{n\rightarrow\infty}(x_{n}-\sqrt{n})=\frac{1}{2}$ where $x_{n}=\sqrt{n+\sqrt{n-1 ...\sqrt{2+\sqrt{1}}}}.$ I've had a lot of ...
0
votes
2answers
35 views

Inequality error possibly. How are two inequalities equal?

Notation: $\underline{x}\in \Bbb R^n,||\cdot||_p =\left(\sum \limits_{i=1}^n |\cdot|^p\right)^{\frac1p}$ $$||\underline{x}||_p\left( \sum \limits_{i=1}^n |x_i + ...
0
votes
2answers
84 views

Integration of $x^n e^{-x} dx$

I've been trying solve this, and even though I feel I'm really close to the answer- I'm quite unsure of the actual answer. The question is a definite integral $$\int_{0}^{\infty} \frac {x^n} ...
0
votes
1answer
16 views

Limits of coefficeints in a system of O.D.E.s.

Say that we have a system of O.D.E.s that depend on some real parameter $c$ \begin{equation*} \dot{x}_i = f_i(c,x_1,...,x_n) \ ,\ \ \ i=1,2,...,n \ . \end{equation*} I've not really seen what it ...
0
votes
3answers
44 views

An integrable and periodic function f(x)

For a periodic function we have: $$\int_{b}^{b+a}f(t)dt = \int_{b}^{na}f(t)dt+\int_{na}^{b+a}f(t)dt = \int_{b+a}^{(n+1)a}f(t)dt+\int_{an}^{b+a}f(t)dt = \int_{na}^{(n+1)a}f(t)dt = \int_{0}^{a}f(t)dt.$$ ...
1
vote
2answers
17 views

If $f(x) \le \mu$ for all $x$ in $S$ and $x_0$ is a limit point of $S$ at which $f$ is continuous, then $f(x_0) \le \mu$.

I'm really lost here and not sure how to use the information given in the problem about $x_0$ being a limit point. Please help.
2
votes
0answers
23 views

Existence of a differentiable function given a unit gradient field

I'm trying to prove that "Given a unit vector field $V$, it can always be uniquely determined a differentiable function $f$ that satisfies $\nabla f = V$." To provide you more information, the unit ...
0
votes
1answer
37 views

How to find out b's value in e^(bx)?

It is killing me......I can not get this right! The answer is A. Can anyone please help!
2
votes
3answers
175 views

Where did I go wrong when doing this integral?

This is my integral $$ \int \frac{ (2x-3)}{(x^3 +10x)}\cdot dx \\ $$ This is my work $$ \int\frac{2x}{x^3 +10x}\cdot dx-\int\frac{3}{x^3 +10x}\cdot dx\\ $$ Looking at them separately: $$ ...
1
vote
2answers
26 views

How to prove this function has a horizontal tangent line at x=0?

By solving the derivative of the function we get two solutions. But do we prove that the third solution is at x=0? The answer is D.
1
vote
1answer
26 views

How to show that if $f$ is a solution for $y" + y = 0$ and the graph of $f$ passes contains 2 arbitrary points then f is unique

Suppose that $f$ satisfies $$y'' + y = 0 $$ And the graph of $f$ contains the points $(a_{1},b_{1})$ and $(a_{2},b_{2})$, with $$a_{1}-a_{2} \neq n \pi, n \in \mathbb{Z}$$ Show that $f$ is the ...
1
vote
1answer
31 views

how to understand Taylor's inequality intuitively?

I am learning the Taylor Series at the moment and I am trying to figure out how to understand Taylor's inequality intuitively. I know you can integrate repeatedly and prove the inequality is ...