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

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0
votes
1answer
14 views

Does $\lim_{x\to \infty}f(x)$ exist?

For any $x\in \Bbb R$, $h(x)\le f(x)\le g(x)$ and $\displaystyle\lim_{n\to \infty}(g(x)-h(x))=0$. Then does $\displaystyle\lim_{x\to \infty}f(x)$ exist? Thanks for your help.
1
vote
2answers
53 views

Differentiating $\left(1āˆ’\frac{1}{x}\right)^{x^2}$

I asked a question about how to differentiate $(1āˆ’1/x)^x$ before, for $x>1$. The derivative I was told is $$f'(x)=(1-(1/x))^x[(1/(1-x))+\log(1-(1/x))]$$ However, this is negative for many ...
-2
votes
2answers
46 views

How do I take the 100th derivative of a polynomial [on hold]

How could I find $$f^{100}(x)$$ for $$f(x)=2x^{100}-7x^{80}+15x^{60}-27x^{40}-18x^{20}+300$$
1
vote
2answers
38 views

Limit radius of convergence $ S = \sum^\infty_{n=0} \frac{(n + 1)!}{8^n} $

Here's the problem as given: Let: $$ S = \sum^\infty_{n=0} \frac{(n + 1)!}{8^n} $$ If I use the Ratio Test to determine whether S converges, I need to determine: $$ \lim_{n\to\infty} ...
6
votes
3answers
120 views

Calculating a limit of integral

Computing the limit: $$\lim_{n\rightarrow\infty}\left(\frac{1}{3\pi}\int_\pi^{2\pi}\frac{x}{\arctan(nx)} \ dx\right)^n$$ I made the substiution $t=nx$ then, we have: ...
-1
votes
3answers
40 views

Limit of a function to the power of another function

Is there a theorem in real analysis for $\underset{n\rightarrow \infty}\lim f(n)^{g(n)}$, where $f(n)$ and $g(n)$ are arbitrary functions of $n$? Under what conditions on $f(n)$ and $g(n)$ does the ...
-4
votes
2answers
39 views

Consider the function with its first and second derivative help please [duplicate]

$$f(x)=\frac{4x^2}{x^2+3} $$ $$f'(x)=\frac{24x}{(x^2+3)^2} $$ $$f''(x)=\frac{72(1-x^2)}{(x^2+3)^3}$$ a)What are the critical numbers(if any)? b)On what intervals is the function increasing and on ...
-4
votes
0answers
29 views

Function continuous at end-points [on hold]

If we have a function $f$ that is absolutely continuous on $(-1,1)$ (and also the derivative of $f$ is absolutely continuous on $(-1,1)$) and we have that for the derivative the following limits ...
2
votes
3answers
133 views

a limit property at infinity

Let $k\in(0,1)$ is fixed and $L$ is a finite value. Is it possible to say if $\lim_{x\to\infty}f(x)=L$ then $\lim_{x\to\infty}f(kx)=L.$
0
votes
0answers
36 views

sum of power series

power series of function I am still working on this series now instead of $\frac{k}{k+1}$ i am taking any arbitrarily sequence of ($x_k$). Here ($x_k$) is convergent sequence. So far i could find ...
0
votes
3answers
41 views

Proving by Cauchy's definition $\lim_{x\to -1} x^2+3x-5=-7$

Prove by Cauchy's definition $\displaystyle\lim_{x\to -1} x^2+3x-5=-7$ From definition: $|x+1|<\delta\Rightarrow |x^2+3x+2|<\epsilon \iff |x+1||x+2|<\epsilon$. Now I'm not really sure ...
3
votes
1answer
53 views

Show that series converge or diverge

If $\displaystyle \sum_{n=1}^{\infty} a_n$ converge and has positive terms then decide if following series converge or diverge : a) $\displaystyle \sum_{n=1}^{\infty} a_n \cdot \sin{a_n}$ I think it ...
2
votes
1answer
40 views

20th derivative of a rational function

I could not find the 20th derivative of the function below : $$f(x) = \frac{2x}{x^2 - 4}$$ I have taken 1st and 2nd derivatives but I could not succeed at generalizing the derivative function.
2
votes
1answer
28 views

Proving by Cauchy's definition $\lim_{x\to 0} x^2\cos x=0$

Prove by definition that $$\displaystyle\lim_{x\to 0} x^2\cos x=0$$ So take $\delta=\sqrt\epsilon$, and from definition we have: $|x|<\delta\Rightarrow|x^2|<\delta^2\Rightarrow|x^2\cos ...
1
vote
3answers
66 views

Why does $1+p+p^2+\dotsb+p^{n-1}=\frac{1-p^n}{1-p}$ [duplicate]

$$y_n=\rho^ny_0+(1+\rho+\rho^2+\cdots+\rho^{n-1})b.$$ If $\rho \not=1$, we can write this solution in the more compact form $$y_n=\rho^ny_0+\frac{1-\rho^n}{1-\rho}b.$$ This is from Elem. Diff. ...
-1
votes
1answer
46 views

Consider the function, f and its second derivative:

$$f(x)=\frac{4x^2}{x^2+3} $$ $$f'(x)=\frac{24x}{(x^2+3)^2} $$ $$f''(x)=\frac{72(1-x^2)}{(x^2+3)}$$ a)What are the critical numbers(if any)? b)On what intervals is the function increasing and on ...
2
votes
2answers
50 views

Proving $\lim_{x\to9}\sqrt x=3$ using Cauchy's definition

Prove: $\displaystyle\lim_{x\to9}\sqrt x=3$ using Cauchy's definition for a limit. After doing the scratch work I get that: $\delta=\epsilon^2+6\epsilon$, so going back, I have to show that ...
5
votes
1answer
99 views

Evaluate $\int \ln(1 + e^x)\ \mathrm dx$

Evaluate the following indefinite integral. $$\int\ln(1 + e^x) \mathrm dx$$ My attempt :: Using integration by-parts, \begin{align} \int\ln(1 + e^x)\cdot 1\ \mathrm dx &= x\ln(1 + e^x) - \int ...
-1
votes
1answer
60 views

Decide convergence of the series .

I have problem with these two: a) $\displaystyle \sum_{n=2}^{\infty} \frac{1}{(\ln{\ln{n}})^{\ln{n}}}$ b) $\displaystyle \sum_{n=3}^{\infty} \frac{1}{n \cdot \ln{n} \cdot \ln{\ln{n}}}$ My try: a) ...
0
votes
4answers
36 views

Why does $\lim_{n \to \infty} \sum_{k=1}^n\frac{t^{k+1}}{(k+1)!}=e^t-t-1$?

Why does $\lim_{n \to \infty} \sum_{k=1}^n\frac{t^{k+1}}{(k+1)!}=e^t-t-1$? I know $\lim_{n \to \infty} \sum_{k=0}^n\frac{t^k}{k!}=e^t$, but my sum starts at $k=1$ and also has ...
3
votes
2answers
38 views

Convergence of a integral: $\int_{0}^{1} |\ln (x)|^n \ dx$

Let $n \in \mathbb N$ be arbitrary. Does the integral $$\int_{0}^{1} |\ln (x)|^n \, dx$$ converge? I asked myself this question and I have no idea of a proof or counter example. Someone can give me a ...
1
vote
1answer
24 views

Find all parameters for for which the series is convergent - checking

I'm not sure if my reasoning is good. Find all parameters $a$ for for which the series is convergent $\displaystyle \sum_{n=1}^{\infty}a^{w_n}$ where $\displaystyle w_n=(\sqrt[n]{2}-1)^{(-1)}$ My ...
1
vote
0answers
28 views

The converse of Lagrange mean value theorem

Let $f$ be a continuous function in the interval $[a,b]$ and differentiable in $(a,b)$. Is it possible, for each $x \in (a,b)$ to find $a_x \in (a,b)$ and $b_x \in (a,b)$ so that $f'(x) = ...
0
votes
0answers
30 views

Generalization of N-Body Problem

I know the n-body problem has been solved for gravity, but in a purely mathematical sense, has it been solved? Or could it be generalized to any kind of field? Maybe an example will make my question ...
0
votes
0answers
46 views

Find $ \int_{\theta_0}^{\theta} \cos \theta \left( \sin 2\theta \right)^{3/2} \, \mathrm{d}\theta $ [on hold]

Find $$ \displaystyle\int_{\theta_0}^{\theta} \cos \phi \left( \sin 2\phi \right)^{3/2} \, \mathrm{d}\phi $$
1
vote
4answers
68 views

The sequence of improper integrals of the form $\int\frac{dx}{1+x^{2n}}$

Let $n\in\mathbb N$ ($n>0$), and define the $n$th integral in the sequence $I$ to be $$I_n = \int_{-\infty}^{\infty}\frac{1}{1+x^{2n}}dx.$$ Evaluating such integrals, especially for small $n$, is ...
0
votes
0answers
27 views

Matrix Derivation involving Chain Rule

I have the following definitions. I have a vector given by $z_{t} = (\alpha_{1t},\ldots, \alpha_{Jt}, \beta_{1t}, \ldots, \beta_{Jt})^{\mathsf{T}}$, where each entry is a scalar value and $t = ...
3
votes
5answers
82 views

Complex Analysis book including integration

FOR BEGINNERS: Currently, I am looking for a textbook on complex analysis, which covers complex analysis from the beginning, and majorly focuses on contour integration, and the residue theorem. On ...
0
votes
0answers
21 views

How do I know if a given polynomial is a quasi polynomial?

How do I know if a given polynomial is a quasi polynomial? For example, if I'm given the polynomial: $e^x\tan(x)$ or the polynomial $e^{(i-t)}t^3$, my gut feeling is that they're both not quasi ...
1
vote
2answers
37 views

Does $\displaystyle\liminf_{n\to \infty} -a_{n}= -\displaystyle\limsup_{n\to \infty}a_{n}$?

Let $(a_{n})$ be a bounded sequence. How to prove $$\displaystyle\liminf_{n\to \infty} -a_{n}= -\displaystyle\limsup_{n\to \infty}a_{n}$$ I don't how formally prove this..can someone guide me? tnx!
3
votes
2answers
27 views

Laplace transform of $f(t)=te^{-t}\sin(2t)$

I was asked to find the laplace transform of the function $f(t)=te^{-t}\sin(2t)$ using only the properties of laplace transform, meaning, use clever tricks and the table shown at ...
0
votes
1answer
27 views

2 question about supremum of subset and a sequence that converge to it.

Let $A$ be a bounded subset of $\mathbb{R}$. 1. Show that there exists a sequence $a_n$ of elements of $A$ such that $\lim _{ }\left(a_n\right)\:=\:sup\left(A\right)$ 2. Show that we can build a ...
3
votes
2answers
98 views

Evaluation of $\int \frac{\sqrt{1+x^4}}{1-x^4}dx$ [duplicate]

Evaluation of $\displaystyle \int \frac{\sqrt{1+x^4}}{1-x^4}dx$ $\bf{My\; Try::}$ Given $\displaystyle \int\frac{\sqrt{1+x^4}}{1-x^4}dx\;,$ Then We can write the above Integral as $$\displaystyle ...
2
votes
1answer
72 views

How to evaluate this indefinite integral $\int\frac{\cos(x)}{1+\mathrm{e}^x}\mathrm{d}x$ [duplicate]

One of my student asked me to help her evaluate this indefinite integral $$\int\dfrac{\cos x}{1+e^x}\mathrm{d}x,$$ and I tried several minutes, but at last I had to given up, for I thought that it is ...
5
votes
2answers
63 views

Quadratic Expressions: Advanced techniques of Integration

$$\int \frac{x}{\sqrt{5+12x-9x^2}}\,dx$$ After two steps I arrive at $\displaystyle{ \int \frac{x}{\sqrt{9-(3x-2)^2}}}\,dx$ Using trigonometric substitution, we have a triangle with a cosine of ...
1
vote
1answer
41 views

Advanced Integration techniques: Quadratic Expressions and U-Substitution

Find $$\int \frac{2x-1}{x^2-6x+13}dx $$ In the final steps after a u-substitution, one arrives at $$\int \frac{2u}{u^2+4}du + \int\frac{5}{ u^2+4}du$$ The next step is arriving at $$\ln(u^2+4) + ...
1
vote
1answer
25 views

Finding pathline

I've been trying to find the pathline of a particle dropped in a steady flow defined by the following vector components: $$ u= \frac{-2x}{(x^2+y^2+1)^2} \hat i + \frac{-2y}{(x^2+y^2+1)^2}\hat j $$ in ...
1
vote
2answers
51 views

Solutions to the integral $\int \frac {dx}{2\sqrt x (x+1)}$

I am given a question to solve the integral $\int \frac {dx}{2\sqrt x (x+1)}$. When I substitute $x+1 = t^2$, I get the solution as $\space \ln(\sqrt{x+1} + \sqrt x) +C$; while when I substitute ...
1
vote
3answers
56 views

A quadratic polynomial is nonnegative for all $x$ if and only if the discriminant is nonpositive

Show that if $a>0$ the inequality $ax^2+2bx+c\ge 0 $ for all values of $x$ if and only if $b^2-ac\le 0$. I tried to prove it by: $ax^2+2bx+cā‰„ b^2-ac$. Used partial derivatives with respect to ...
0
votes
1answer
49 views

differential inequality implies zero function

Let $f$ be twice continuously differentiable on $(-1,1)$, and $f(0)=f'(0)=0$, $\quad|f''(x)|\leq |f(x)|+|f'(x)|$. Show that $f=0$ in some neighborhood of $0$. How can we deduce something from this ...
0
votes
2answers
29 views

Evaluation of an integral of some expressions involving fractions

I am stuck in evaluating the following integral: \begin{equation} \int_{0}^{b-a} \frac{1}{\sqrt{u} (a+u)} \,du, \end{equation} where $0<a<b$. Any ideas?
1
vote
1answer
27 views

Mean value inequality geometrical interpretaion

The mean value inequality theorem Let U be an open interval in $\mathbb{R}$. Suppose that $K \ge 0$ and that, $a,b \in U$ with $b>a$. If $f : U \rightarrow \mathbb{R}$ is differentiable with ...
5
votes
2answers
32 views

Integration of high order fraction; explanation of method.

I understand the cases where the order is less or equal to 3 (example where it is three we split numerator with A,B,C), but in this case(Example 8) I do not see why we split the numerator with A, ...
2
votes
1answer
24 views

$k_{n+1}\le (1+2\varepsilon)k_n$ for $k_n:=\lfloor(1+\varepsilon)^n\rfloor$ and $\varepsilon>0$

Let $$k_n:=\lfloor(1+\varepsilon)^n\rfloor\stackrel{\text{def}}{=}\max\left\{k\in\mathbb{Z}:k\le(1+\varepsilon)^n\right\}\;\;\;\text{for }n\in\mathbb{N}$$ How can we prove $k_{n+1}\le ...
0
votes
0answers
21 views

Fourier series - Understanding an equality

Why is this equality true: $$\left\langle {f,g} \right\rangle = \sum\limits_{n = - N}^N {\hat{f}(n)\hat{g}(n)}$$ where $$f = \sum_{n=-N}^N c_n e^{int}, g=\sum_{n=-N}^N d_n e^{int} $$ and ...
0
votes
1answer
38 views

Using the Maclaurin series to approximate $f(0.1)$ for $f(x)=(3+e^{2x})^{0.5}$

I was tasked to use the Maclaurin series to calculate $f(0.1)$ of $f(x)=(3+e^{2x})^{0.5}$. I got the Maclaurin expansion of $p_2(x) = \sqrt{3} + 4x +5x^2$ into which I plugged $0.1$ to yield ...
1
vote
0answers
35 views

What's wrong with my reasoning while setting up a limit?

I was writing an answer to this question, which asks about what happens to the apex of an isosceles triangle if a vertex is at infinity. I thought it would be very easy to prove it by setting up a ...
0
votes
2answers
35 views

Proof for pythagoras theorem

Let $f,g$ orthogonals to each other. $${\left\| {f + g} \right\|^2} = \left<f,f\right>+\left<g,f\right>+\left<f,g\right>+\left<g,g\right> = {\left\| f \right\|^2} + {\left\| g ...
1
vote
0answers
27 views

A treatise on Probabilistic arguments and Laplace/Fourier transforms to solve limits/integrals from basic calculus.

I've seen in some answers in Brilliant.org to some very complicated limits and integrals that uses probabilistic arguments (Let $X$ be a random variable from $[0,1]$... some examples are in those ...
5
votes
4answers
164 views

Finding $ \int_0^1 \frac {\ln x}{1+x^2}\mathrm dx $

Today I encountered the problem of how to find $$ \displaystyle\int_{0}^{1} \frac {\ln x}{1 + x^2}\mathrm dx $$ but got no start on it. Is this one of those integrals which we have to approach from ...