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

learn more… | top users | synonyms

4
votes
0answers
40 views

Questionable Convergence of a Series

The summation is: $$ S = \sum_{k \geq 0} f(k) \int_{0}^{\pi/2} \sqrt{1-(1- \frac{f(k+1)^2}{f(k)^2})\sin^2(\theta)}d\theta $$ Now, we know that $f(k+1) < f(k)$ and as $k$ approaches infinity, ...
0
votes
1answer
47 views

Applicability of tests of convergence for series with non-negative terms

We know that there are many criteria of convergence for series with non-negative terms (for example, ratio test (with limit), root test (with limit), integral, comparison, and asymptotic comparison, ...
8
votes
2answers
506 views

Compute $\sum\limits_{n = 1}^{\infty} \frac{1}{n^4}$.

Compute the Fourier series for $x^3$ and use it to compute the value of $\sum\limits_{n = 1}^{\infty} \frac{1}{n^4}$. I determined the coefficients of the Fourier series, which are $$a_0 = ...
2
votes
4answers
75 views

The constant of integration in the solution to the differential equation $-4 g(x)=2 x g'(x)$

When I solved this differential equation--- $$-4 g(x)=2 x g'(x)$$ ---I obtained $$\log (g(x))=-2 \log (x).$$ Solving for g(x) I got $\frac{1}{x^2}$. Now this is an error that I constantly ...
1
vote
1answer
83 views

For which values of $q$ and $\alpha$ does $\sum_{n=0}^\infty q^{n^\alpha}$ converge?

Let us consider the series $$\sum_{n=0}^\infty q^{n^\alpha}, \qquad \qquad\qquad (*)$$ with $\alpha,\in \mathbb{Q}, n \in \mathbb{N}$. I know that the geometric series converges if $|q|<1$, but ...
0
votes
2answers
124 views

Prove limit doesn't exist using epsilon-delta defintion

Prove $\lim_{x \to 0}\frac{x}{x-\lfloor \sin x \rfloor}$ doesn't exist. I can prove the same claim using Heine's definition by showing that sequences $\{x_n\}=\frac{\pi}{2n}$ and $\{x_m\} = ...
0
votes
0answers
37 views

Gradient of a function at the boundary of a constant region

Seemingly an easy thing to do, I had difficulty to find an answer for the following: Let's assume we have a function $f(x)$ which is defined as $f:\mathbb{R^n} \to \mathbb{R}$. The function has ...
-6
votes
1answer
74 views
1
vote
1answer
20 views

define $f(\mathbf{x})=f_1(x_1)+\cdots +f_n(x_n)$. Show that $f$ has a differential at each point of an n-dimensional interval.

Given $n$ real-valued functions $f_1, \dots, f_n$, defined and having finite derivatives in the interval $(a,b)$. For each $\mathbf{x}$ in the $n$-dimensional interval $$S=\{(x_1,\dots ,x_n)\mid a\lt ...
2
votes
0answers
96 views

Why is the negative of the gradient the direction of greatest descent?

I imagine it as if one is going up a physical hill. It doesn't seem like there's a guarantee that going in the opposite direction of greatest increase in height will necessarily be the direction of ...
2
votes
1answer
62 views

How do I differentiate a Kronecker product with respect to a vector?

I am trying to differentiate $[\mathbf{I} \otimes \mathbf{t}^*\mathbf{t}^T]$ with respect to $\mathbf{t}$. I did the following $\mathbf{I} \otimes \mathbf{t}^*\mathbf{t}^T = (\mathbf{I} \otimes ...
1
vote
1answer
42 views

Taylor Polynomial And Reminder In Lagrange Form

What is the error of Taylor polynomial of $\sin(x)$ of order 4 around $x_0=0$ at the point $-1$? So the Taylor polynomial is $x-\frac{x^3}{3!}$ and the remainder in Lagrange form is ...
0
votes
1answer
182 views

Convergence of $\sum_{n=0}^\infty \frac{\sqrt{1+x^n}}{x^n}$ in the case $x<0$ and an analogous problem with $\sum_{n=0}^\infty \frac{x^n}{2+x^n}$

Let $$\sum_{n=0}^\infty \frac{\sqrt{1+x^n}}{x^n}.$$ My first question is: for what values of $x$ is this series possible? I can only say that it is not defined for $x = 0$, but are there other ...
4
votes
1answer
220 views

Analyzing the convergence of an improper integral

I have to analyze the convergence of $$\int _{0}^{+\infty} \frac{\cos \left( x\right) -1} {x^{5 / 2}+5x^{3}}\,dx$$ I've rewritten the integral as $$ \int _{0}^{+\infty} \frac{\cos \left( x\right) ...
0
votes
1answer
69 views

laplacian of $1/\rho$ in cylindrical coordinates

In spherical coordinates, I believe that the laplacian of $1/r$ is zero everywhere except at $r = 0$ or \begin{align} \nabla^2 \dfrac{1}{r} = -4\pi \delta^{(3)}({\vec{r}}). \end{align} where $r$ is ...
6
votes
1answer
254 views

Why is the Cauchy product of two convergent (but not absolutely) series either convergent or indeterminate (but does not converge to infinity)?

It is well-known that the Cauchy product of two absolutely convergent series is absolutely convergent. However, my professor added (without giving a proof) that if the series are convergent ...
2
votes
3answers
2k views

How do you show that $d\theta = \frac{x dy - y dx }{x^2 + y^2}$?

If $(r, \theta)$ are polar coordinates on $\mathbb{R}^2\setminus \{ (0,0)\}$, then how do I show/prove that \begin{equation*} d\theta =\dfrac{x dy - y dx}{x^2 + y^2}? \end{equation*}
2
votes
3answers
93 views

Does $\sum_{n=2}^\infty\frac{1}{n\ln n!}$ converge?

Let $$\sum_{n=2}^\infty \frac{1}{n\ln n!}.$$ It is equal to $$\sum_{n=2}^\infty \frac{1}{n(\ln n + \ln(n-1) +...+ ln(2))}.$$ But now what should I do to prove that it converges? (I have tried root ...
0
votes
2answers
51 views

Which of the statments are true?

Which of the statments are true? $1.$ if $\lim_{x\rightarrow a}(f(x)+g(x))=0$ and $\lim_{x\rightarrow a}f(x)=3$ then $\lim_{x\rightarrow a}g(x)=-3$ $2.$ if $\lim_{x\rightarrow \infty ...
3
votes
1answer
105 views

Solving an equation including $e^{-x}$ with the Lambert W function

Given two functions of $x$, namely $f(x)$ and $g(x)$, where $$f(x)=x^2-4x+8$$$$g(x)=3xe^{-x}$$ the shortest distance between the graphs of the functions is sought. I begin by defining a function ...
1
vote
1answer
65 views

Calculating cardinality of the following sets

I want to calculate the cardinality of the various sets such as: The set of continuous functions from $\mathbb R$ to $\mathbb R$. The set of continuous functions from $\mathbb Q$ to $\mathbb Q$ The ...
0
votes
2answers
42 views

Study: $\sum_{n=1}^\infty (\sin(\sin n))^n$, $\sum_{n=1}^\infty \frac{n \sin (x^n)}{n + x^{2n}}$, and $\sum_{n=1}^\infty \frac{ \sin (x^n)}{(1+x)^n} $

Let $x \in \mathbb{R}$. I have to study the convergence of the following three series: $$\sum_{n=1}^\infty (\sin(\sin n))^n$$ $$\sum_{n=1}^\infty \frac{n \sin (x^n)}{n + x^{2n}}$$ ...
4
votes
2answers
5k views

How do you solve this series question? : $\cos (n\pi )/ \ln(6n) $

The problem is, Select the FIRST correct reason on the list why the given series converges. A. Geometric series B. Comparison with a convergent p series C. Integral test D. Ratio test E. ...
0
votes
1answer
37 views

Calculate the sum of the series $\sum_{n=0}^{\infty} \frac{1}{a_{n}\cdot a_{n+1}\cdot\ldots\cdot a_{n+7}}$, where $a_k = ak + b$

Let $a_k = ak + b$; define the following series: $$\sum_{n=0}^{\infty} \frac{1}{a_{n}\cdot a_{n+1}\cdot\ldots\cdot a_{n+7}}.$$ I have to prove that this series converges and I have to find its sum. ...
3
votes
0answers
82 views

Fast Hankel Transform [closed]

Can someone please explain what would be the expression for weights(Ho) in a Fast Hankel Transform.I found this in a paper and could not find any satisfactory answers .
0
votes
1answer
18 views

When should I use total differentials and when should I use implicit differentiation?

When should I use total differentials and when should I use implicit differentiation? An example of an equation where I'm unsure if to use total differentials or implicit differentiation is $f(x, y) ...
6
votes
1answer
88 views

A limit with the harmonic series

How can we prove the following (similar) limits? $$\sum_{k=1}^n \frac{1}{k} (\ln 2 - \frac{1}{n+2} - \frac{1}{n+3} - \cdots -\frac{1}{2n + 2}) \to 0. $$ $$\sum_{k=1}^n \frac{1}{k} (\ln 3 - ...
0
votes
1answer
23 views

Balancing the area of curves - integration

A linear equation needs to be found $y=ax+b$ with a slope of a maximum value of $10$ degrees that will balance out the area above the line and below the line for the function ...
5
votes
2answers
283 views

Why do Lagrange Multipliers work?

I know that the Lagrange multiplier method helps us evaluate critical points of $f$ on the closed boundary of the restriction. In other words we solve:$$\nabla f=\lambda \nabla g$$ But why does ...
1
vote
4answers
131 views

Finding $\displaystyle \lim_{n \to \infty} \frac{2^{n+1} + 3^{n+1}}{2^{n}+3^{n}}$

I need help with finding $\displaystyle \lim_{n \to \infty} \frac{2^{n+1} + 3^{n+1}}{2^{n}+3^{n}}$ Thanks!
2
votes
2answers
73 views

Continuity of $\frac{1}{|x|}$ at $x= 0$

The function $|x|$ is continuous at zero. What can I say about the continuity of $\frac{1}{|x|}$ ? I have two counter arguments for it continuity. Please suggest what is right. The function is not ...
22
votes
6answers
945 views

A limit problem $\lim\limits_{x \to 0}\frac{x\sin(\sin x) - \sin^{2}x}{x^{6}}$

This is a problem from "A Course of Pure Mathematics" by G H Hardy. Find the limit $$\lim_{x \to 0}\frac{x\sin(\sin x) - \sin^{2}x}{x^{6}}$$ I had solved it long back (solution presented in my blog ...
11
votes
7answers
6k views

How to solve an exponential equation with two different bases: $3^x - 2^x = 5$

Can anyone tell me how to solve this equation $$3^x - 2^x = 5$$ other than graphically? I'm stunned. I don't know what to do in the first step.
0
votes
2answers
49 views

Calculus Summations

Is there any way I can change a summation, say from $k=1$ to $n$ of the derivative of order $k$ of a function into closed form, or some form that would be more manageable? Ex. From $k=1$ to $3$ of ...
0
votes
1answer
23 views

fourier series notation question

Find the fourier series for the given function $$f(x)=-x \quad \text{for } -L\le x < L, f(x+2L)=f(x)$$ this is a question from my book, and im just wondering about one thing and that is what does ...
4
votes
5answers
1k views

Don't understand the Fundamental Theorem of Calculus

If $f$ is continuous on $[a, b]$ and defining $$ F(x) = \int_a^x \! f(t) \, dt $$ for $x \in [a, b]$, then $F'(x) = f(x)$ for $x \in (a,b)$. I don't understand what function the variable ...
3
votes
0answers
59 views

Strange triple integral of an inverse function

Let $$ \Omega(a, b, c) = \min\left\{\theta\ge0\ \text{s.t.}\ \tan(a\theta) + \tan(b\theta) + \tan(c\theta) = 1\right\} $$ What is the value of the following integral $$ I = ...
1
vote
1answer
56 views

How to integrate hydrostatic force on a two dimensional shape?

I'm so confused this question is very different from the other hydrostatic force questions and I think I am misunderstanding the question. I am primarily concerned with 15 because I somehow managed ...
2
votes
2answers
107 views

Converting a word problem into an equation? Trignometry and calculus [duplicate]

The problem is question 3 of what I am about to download into this question. I drew a diagram of what the problem actually is, my professor has verified it's correct. I don't want an exact answer to ...
0
votes
1answer
47 views

Write down the Maclaurin series for the function.

(d) Another function is defined on the interval of convergence of the power series in (b) by the formula $\displaystyle{g(x)=\int_0^x \ln(2t+1) dt}$. Write down the Maclaurin series for the ...
2
votes
0answers
49 views

For which values of $\alpha, \beta, x \in \mathbb{R}, x \geq 0$, does the series $\sum_{n=1}^\infty n^ \alpha x^{n^{\beta}}$ converge?

I have to study, for $\alpha, \beta, x \in \mathbb{R}$, $x \geq 0$, the convergence/divergence/irregularity (i.e., when the limit of the $N$-th partial sum does not exist) of the following series: ...
2
votes
4answers
116 views

Sufficient condition for convexity

Let f:$ [a,b] \rightarrow \mathbb{R} $ a continous function such that $ \forall (x,y) \in [a,b]^{2}, \exists t \in ]0,1[, f(tx+(1-t)y) \le tf(x) + (1-t)f(y) $ show that f is convex
1
vote
2answers
82 views

For which $x \in \mathbb{R}$ does the series $\sum_{n=1}^\infty \frac{x^n}{n!}$ converge?

One problem of my exercise book asks for which $x \in \mathbb{R}$ the following series converges: $$\sum_{n=1}^\infty \frac{x^n}{n!}.$$ The answer given by the exercise book is $|x|\leq 1$, but ...
4
votes
1answer
790 views

Geometrical Interpertation of Cauchy's Mean Value Theorem

Cauchy MVT: If functions f and g are both continuous on the closed interval [a,b], and differentiable on the open interval (a, b), then there exists some c ∈ (a,b), such that $$\frac{f'(c)}{g'(c)}= ...
2
votes
1answer
881 views

General solution of $(x^2-y^2)dx + (3xy)dy = 0$

Find the general solution to the homogeneous differential equation $$(x^2-y^2)dx + (3xy)dy = 0$$ The differential equation does not seem to be separable, and I'm having a tough time to put it in the ...
0
votes
1answer
38 views

Find Taylor Polynomial

Find the Taylor Polynomial of $f(x)=\sin(e^{x+2})$ around $x_0=-2$ and of order $2$. We can write $f(x)=\sin(t)$, where $t=e^{x+2}$, so we have: $f(-2)=\sin(e^{x+2})=\sin(e^{0})=\sin(1)$ ...
2
votes
4answers
380 views

Area between three lines/curves

I know this is a very elementary question but I can't make out the answer from the other posts I found in my search. These are three lines, I need to find the area enclosed by them. how do I go ...
6
votes
1answer
57 views

For which $x\in \mathbb{R}$ does $\sum_{n=1}^\infty \left(\frac{x^{2n}}{n} - \frac{n^{2x}}{x}\right)$ converge?

I have to study for which values of $x \in \mathbb{R}$ the following series converges: $$\sum_{n=1}^\infty \left(\frac{x^{2n}}{n} - \frac{n^{2x}}{x}\right)$$ I was only able to say that the ...
0
votes
0answers
31 views

When is system of linear equations smooth

I am wondering when is a system of linear equations smooth? More specifically, for Ax=B, what property of A guarantees smoothness of systems of linear equations? If it is known that A is always ...
2
votes
1answer
96 views

A piecewise $C^1$ curve has Jordan measure zero.

$\newcommand{\Reals}{\mathbb{R}}\gamma:[0,1]\to \Reals^2$ is an injective parametrization of a curve $\Gamma$, which is piecewise $C^1$ and the length of the curve is $L(\Gamma_k)<\infty$. 1.1.: ...