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

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0answers
25 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
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3answers
31 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
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1answer
43 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
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1answer
34 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
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1answer
27 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
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3answers
63 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. ...
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1answer
31 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
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2answers
45 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 ...
4
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1answer
85 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 ...
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1answer
55 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
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4answers
32 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
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2answers
35 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
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1answer
22 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
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0answers
24 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) = ...
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0answers
24 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 ...
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0answers
42 views

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

Find $$ \displaystyle\int_{\theta_0}^{\theta} \cos \phi \left( \sin 2\phi \right)^{3/2} \, \mathrm{d}\phi $$
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4answers
65 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
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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
66 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
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0answers
20 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
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2answers
35 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
26 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
17 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
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2answers
95 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
69 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
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2answers
59 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
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1answer
40 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
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1answer
24 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 ...
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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 ...
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3answers
53 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: $ax^2+2bx+c≥ b^2-ac$. Used partial derivatives with respect to ...
0
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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
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2answers
28 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
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1answer
26 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
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2answers
30 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
23 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
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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
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1answer
37 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
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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
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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
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0answers
26 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
162 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 ...
4
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2answers
93 views

Probabilistic techniques, methods, and ideas in (“undergraduate”) real analysis

As the book Probabilistic Techniques in Analysis by Richard Bass shows, nowadays techniques drawn from probability are used to tackle problems in analysis. The mentioned book presents a survey of ...
2
votes
4answers
54 views

Prove that a function is continuous for every $x \in R$

Prove that the function: $$ f(x)=\frac{\sqrt{x^2-x+1}}{|\sin(x)-4|-2} $$ is defined for every $x \in R$ and continuous in every $x \in R$, So I said that in order for this function to be defined we ...
3
votes
3answers
61 views

Find $ \int \frac {1-x^2}{1+3x^2+x^4} \, \mathrm{d}x $

Today, the CalcBee sample problems got released. The following problem was my creation and I wanted to see how many solutions people can come up with. The result is very beautiful and I thought it ...
5
votes
4answers
118 views

Evaluating $\int{\frac{1}{\sqrt{x^2-1}(x^2+1)}dx}$

Evaluating $$\int{\frac{1}{\sqrt{x^2-1}(x^2+1)}dx}$$ using $ux=\sqrt{x^2-1}$ I try to $u^2x^2=x^2-1$ $x^2=\frac{-1}{u^2-1}$ However I cant get rid of $x$ because derivative has $x\;dx$. How can I ...
4
votes
1answer
24 views

Trigonometic Substitution VS Hyperbolic substitution

The following tables were taken from University of Pennsylvania's page about Calculus: Trigonometric Substitution Hyperbolic Substitution As you can see, the forms $1+x^2$ and $x^2-1$ are repeated ...
1
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4answers
40 views

Calculate a limit of exponential function

Calculate this limit: $$ \lim_{x \to \infty } = \left(\frac{1}{5} + \frac{1}{5x}\right)^{\frac{x}{5}} $$ I did this: $$ ...
5
votes
2answers
93 views

Closed form of $\int_0^1\int_0^1\int_0^1\frac{\left(1-x^y\right)\left(1-x^z\right)\ln x}{(1-x)^3}\,\mathrm dx\;\mathrm dy\;\mathrm dz$

While trying to find several references to answer Pranav's problem, I encounter the following multiple integrals $$I=\int_0^1\int_0^1\int_0^1\frac{\left(1-x^y\right)\left(1-x^z\right)\ln ...
0
votes
1answer
36 views

sequence defined by norm

Let $(u_n)_n$ be defined by: $\quad \begin{cases}u_1=1 & \\ \\ u_n=\left( \sum\limits_{k=1}^{n}u_{k}\right)^{\frac{1}{2}} & \end{cases} $ Show that $u_{n}\to +\infty$ and $u_n \simeq ...
1
vote
0answers
88 views

Evaluation of $\displaystyle\int\frac{1}{x^4+1}$ (Spivak's Calculus, Chapter 19, Problem 6viii) [duplicate]

Solve the integral:$$\int \frac{1}{x^4+1}\mathrm{d}x$$ I tried the substitution $x=\tan\theta \Rightarrow dx=\sec^2 \theta\,\mathrm{d}\theta$, but that leads to a dead end. $$\begin{align}\int ...