Mathematical analysis. Consider a more specific tag instead: (real-analysis), (complex-analysis), (functional-analysis), (fourier-analysis), (measure-theory), (calculus-of-variations), etc. For data analysis, use (data-analysis).

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0
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3answers
60 views

How to find a derivative of $f(x)=\int_0^{x^2}e^{xt^{-2}}dt$

Let $$f(x)=\int_0^{x^2}e^{xt^{-2}}dt$$ I want to find $$f'(x)$$ I tried to use taylor expansion: $$e^{xt^{-2}}=\sum_{n=0}^\infty \frac {x^nt^{-2n}} {n!}$$ Indefinite integral gives, $$\int e^{xt^{-2}}...
1
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2answers
84 views

Prove or disprove: $\sup\left \{ x\in\mathbb{R}\mid x^2-5x+6\leq0 \right \}=3$

No homework:http://www2.mathematik.hu-berlin.de/~gaggle/S15/MATHINFO/UEBUNG/nachholklausur.pdf Prove or disprove: $\sup\left \{ x\in\mathbb{R}\mid x^2-5x+6\leq0 \right\} =3$ I would say the ...
1
vote
1answer
31 views

Prove that a complex series diverges.

I am pretty rusty on convergence tricks. I want to check why the series $$ \sum_{k=1}^{\infty} \frac{1}{k+i}, $$ diverges. This is something like the harmonic series but for complex numbers. One way ...
1
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0answers
21 views

If $f(x)\geq 0$ for all $x \in [a,b]$ and $\alpha \in BV([a,b])$ is increasing , then $\int_a^bf d\alpha \geq 0.$

This is a proof verification question. Here, $\, f$ is continuous and $\alpha$ is of bounded variation. My only tools are the sums, for a given partition $P = \{a=x_0 < \ldots < x_n = b \}$ of $...
0
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2answers
30 views

Countable Sets & Infinity

Recently I've been reading "Principles of Mathematical Analysis" by Rudin, and have just begun the section on basic topology. The first theorem it presents is the following: Theorem Every infinite ...
4
votes
1answer
51 views

Baby Rudin - Theorem 1.35 Cauchy Schwartz

I'm stumped on the following difficulty while reading baby Rudin (p.15). Let $A=\sum|a_j|^2, B=\sum|b_j|^2, C=\sum a_j \overline{b}_j$: $$\begin{align} \sum|Ba_j-Cb_j|^2 &= \sum(Ba_j-Cb_j)(B\...
4
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3answers
35 views

Find all local extremums of $f(x)=x^{2}e^{-x}$ and decide if these are global extremums

As all my other questions, this one isn't homework (it's preparation for an exam). I'd like to know if I did everything correctly. In my previous task, I had a mistake in the first derivation. But ...
1
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1answer
37 views

Find all local extremums of $f(x)=\frac{x}{x^{2}+x+1}$ and decide if these are global extremums

Did I do everything correctly? Find all local extremums of the following function and decide if these are global extremums (i.e. maxima or rather minima of the function on its entire domain) ...
3
votes
1answer
30 views

Convergence of a integral for every curve in the sphere

Let $S$ be the unit open sphere in $\mathbb{R}^3$: $x^2+y^2+z^2< 1$ and $\partial S$ its border $x^2+y^2+z^2= 1$. Let $f:S\cup \partial S\rightarrow \mathbb{R}$ be a continuous function which is ...
0
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1answer
23 views

Choice of the limits for multivariable integral

Let $A \subseteq \mathbb{R}^2$ a limited set bordered through $x=0, x=1, y=-1+x, y=1-x^2$. Rotate A around the y-axis and define this set with $B$. Calculate the integral $$\int_B y\,\mathrm{d}x\...
1
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0answers
33 views

Different Alternate Representations of Functions

Could someone please point out a source with detailed steps / other pointers for different alternate representations of functions? For example, I know of two such ways 1) Taylor Series Expansion 2)...
0
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1answer
34 views

Recursive formula in term of original value

$$P_1=P_0G_{0,1}A_1\\ P_m=P_{m-1}G_{m-1,m}A_m+A_0\sum_{i=0}^{m-2}P_i G_{i,m}~~\text{for}~~m\geq 2$$ Is it possible to write $P_m$ in terms of only $P_0$, i.e., without other $P_j$ terms?
2
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2answers
30 views

Calculate $P'(x)$ for $x \in (-1,1)$

$P$ is a power series with $P(x)=x-\frac{1}{3}x^3+\frac{1}{5}x^5-\frac{1}{7}x^7+\cdots$ Calculate $P'(x)$ for $x \in (-1,1)$ When I read this task (not homework), I got some questions: 1) ...
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0answers
85 views

On the nontrivial zeros of the Riemann zeta function. [on hold]

Please kindly forgive me since this topic might not be very suitable for this platform. But i just stumbled upon something on the Riemann zeta function, which made me instantly curious: Observe that, ...
2
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2answers
53 views

Proof that $P(x)=x-\frac{1}3 x^3+\frac{1}{5}x^5-\frac{1}{7}x^7+\cdots$ has radius of convergence $1$

Proof that $P(x)=x-\frac{1}{3}x^3+\frac{1}{5}x^5-\frac{1}{7}x^7+\cdots$ has radius of convergence $1$ First of all, I need to convert this to a series: $$\sum_{k=1}^\infty \frac{x^{2k-1}(-1)^k}{...
1
vote
1answer
26 views

Is the mathematical syntax correct here (Taylor-polynomial)?

Say I'm supposed to create the $2^{nd}$ degree Taylor-polynomial of $f(x) = \cos x$ at $x_{0} = 0$ I'd like to know if the syntax is correct, how I solved this little task. We have defined the ...
0
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1answer
49 views

Does there exist any non-zero polynomial $f:\mathbb C \to \mathbb C$ such that $f(x+2)-2f(x+1)=f(x) , \forall x \in \mathbb C$ ?

Does there exist any non-zero polynomial $f:\mathbb C \to \mathbb C$ such that $f(x+2)-2f(x+1)=f(x) , \forall x \in \mathbb C$ ?
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2answers
16 views

Cauchy Product Radius of Convergence

How do we see that the radius of convergence (RCV) of the Cauchy product is at least the minimum of the two respective RCV's? For instance suppose $f(x)=\sum_{i=0}^\infty a_i(x-x_0)^i$ with RCV $R_1$....
1
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0answers
19 views

statistical comparison, 3 groups, multiple columns

I am using R for some statistical analysis. I have a dataset listing number of deaths by eu regions. the dataset is annual and is for 2000-2008. I divided this data into 4 subgroups according to ...
1
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1answer
28 views

convergence of power serial $\sum_{n=0}^{\infty}\frac{n^{1000}}{\sqrt{n!}}x^n$

$$\sum_{n=0}^{\infty}\frac{n^{1000}}{\sqrt{n!}}x^n$$ I am trying to find such $x$ that this serial is covergent. $$\left|\frac{a_{n+1}}{a_n} \right|= \frac{(n+1)^{1000}}{\sqrt{(n+1)!}} \cdot \frac{\...
3
votes
3answers
69 views

Taylor-polynomial of function $f(x) = e^{x}*\sin(2x)$

This is not homework, I'm asking to learn for an exam which I'll write in 2.5 months. Count the Taylor-polynomial 3th grade of the function $f: \mathbb{R} \rightarrow \mathbb{R}, f(x) = e^{x}*\sin(...
0
votes
2answers
27 views

Let $f:\mathbb{R}^n \to \mathbb{R}$. If $f\in C^1$ then is true that if $Df(a) = 0$ for some $a$ then $Df(x) = 0$ in a neighborhood of $a$?

Let $f:\mathbb{R}^n \to \mathbb{R}$. If $f\in C^1$ then is true that if $Df(a) = 0$ for some $a$ then $Df(x) = 0$ in a neighborhood of $a$? It is strange, for example, $f(x) = \frac{x^3}{3}.$ We have ...
4
votes
1answer
68 views

Vector field and differential equation

We consider from $\mathbb{R}^2$ \ $\left \{ 0 \right \}$ the vector field $$X(x,y)=\left ( \dfrac{x}{x^2+y^2},\dfrac{y}{x^2+y^2} \right )$$ How to show that the differential equation $\dot{\gamma }(...
1
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0answers
35 views

Is this Banach fixpoint iteration proof correct?

I'm preparing myself for an exam and don't have the solution to this task.Maybe you could look through it. We are looking for the zero $x^*$ of $f(x)=\frac{1}{9}x^3-\frac{1}{3}x^2-\frac{1}{2}x+\frac{...
0
votes
2answers
55 views

Taylor-polynomial of $f(x)=\log(\cos(x))$

$f: (-\frac{\pi}{2}, \frac{\pi}{2}) \rightarrow \mathbb{R}, f(x) = \log(\cos x)$ Count the Taylor-polynomial $T_{2}(f, 0)(x)$ of the second degree of $f$ in $x_{0} = 0$ Alright because it was ...
1
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0answers
36 views

The Jeep Problem with Equally Spaced Stations

Consider the following problem. A jeep can carry a maximum load of fuel of 1 gallon, and it travels $l$ miles with $l$ gallons of fuel. The jeep moves along a straight line, and is required to cross a ...
1
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0answers
14 views

Is it possible to use Lusin's Theorem to derive Frechet's Theorem?

Frechet's theorem states that every measurable function $f$ on $\mathbb{R}$ is the limit of a sequence of continuous functions converging almost everywhere. Frechet's theorem is then used to prove ...
0
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0answers
42 views

Rigorious formulation of approximation of integral as square for large 2nd derivative.

We know that the taylor expansion of $$\left(\int_{t}^{t+\Delta t}a(t')dt'\right) = a(t)\Delta t + \frac{1}{2}\frac{da}{dt}(t) \Delta t^2 + \frac{1}{3!}\frac{d^2a}{dt^2}(t) \Delta t^3 + \frac{1}{4!}\...
4
votes
1answer
44 views

If $m^*([-n,n] \cap E) + m^*([-n,n] \setminus E) = 2n$ for all $n$, then $E$ is Lebesgue measurable

Let $E \subset \Bbb R$ and let $m^*$ denote the Lebesgue outer measure on $\Bbb R$. Show that if for all $n \in \Bbb N$, $m^*([-n,n] \cap E) + m^*([-n,n] \setminus E) = 2n$, then $E$ is Lebesgue ...
1
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1answer
76 views

The Jeep Problem and Nash's Friends

The classical jeep problem is the following. A jeep can carry a maximum load of fuel of 1 gallon, and it travels $l$ miles with $l$ gallons of fuel. The jeep moves along a straight line, and is ...
0
votes
1answer
19 views

Dual of the Banach space of $k$-times continuously differentiable functions.

Let $C^k([0,1])$ denote the Banach space of $k$-times continuously differentiable functions $f:[0,1]\to \mathbb R$ with norm $$\|f\|_{C^k}:=\max_{i=0,\dots,k}\sup_{x\in [0,1]}|f^{i}(x)|.$$ I'm trying ...
0
votes
1answer
31 views

Diagram of a multivaribale function

I have to draw the diagram of the function: $$(x^2+y^2)^{\frac{3}{2}}=x^2-y^2$$ I transformed it with polar coordinates to: $$r=\cos^2(\varphi)-\sin^2(\varphi)$$ with $r \ge 0$ and $\cos^2 \ge sin^2$....
2
votes
1answer
36 views

Show that the series $\sum_{k = 0}^{\infty}a_{k}b_{k}x^{k}$ converges absolutely

This is no homework, it's part of a sample exam which can be found here: http://www2.mathematik.hu-berlin.de/~gaggle/S15/MATHINFO/UEBUNG/nachholklausur.pdf Given: Two power series $\sum_{k = 0}^{\...
0
votes
1answer
21 views

Showing properties of a function and its inverse image

I tried proving the following question but did not get too far. Let $\ f:A \to B$ be a function and $\ f^{-1}(Y)$ be the inverse image of $\ Y\subseteq B$ on $\ f$. Consider the following ...
4
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2answers
66 views

Is my proof of $\lim_{x\rightarrow c}x^2=c^2$ correct?

I know the most common proof of $\lim_{x\rightarrow c}x^2=c^2$. But I wonder if my alternative proof is valid and correct. Here's my proof. Let $\varepsilon>0$, want to find a $\delta>0$ such ...
1
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1answer
21 views

Vector function of distance traveled.

Let the scenario be the following: We have a driving car whose start velocity is $100\frac{m}{s}$ and it's brakes reduce the velocity by $10\frac{m}{s}$, quite simple. If we were to make a vector ...
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1answer
28 views

It is about infinity series. [on hold]

$m$ is a given number. Find the sum of $$\sum_{n=1}^\infty\frac{1}{n\times(n+m)}\text.$$
1
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2answers
28 views

Construct a function which has a minimum, two saddle points and no more critical points.

Construct a polynomial $p(x) \in \mathbb{R}[x]$ that the function $f(x,y)=y^2+p(x)$ of $\mathbb{R}^2$ has a minimum, two saddle points and no more critical points. I do not know how to solve it but ...
2
votes
1answer
29 views

Integral of Simple Functions converges to Integral of Measurable Function

Let $f$ be a measurable function and $E_{n,m} = \{x : \frac{m}{2^n} \leq f(x) < \frac{m+1}{2^n} \}$. Prove: $$\lim_{n \to \infty} \sum_{m=1}^{\infty} \frac{m}{2^n} \mu(E_{n,m}) \to \int f \, d\mu$...
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0answers
20 views

uniform convergence of lagrange polynomials , exercise 12.16.15 dieudonne treatise vol 2

This is exercise 12.16.15 from Dieudonne's treatise on analysis volume 2 It attempts to find necessary and sufficient conditions on a sequence of control points in I = [0,1] for the lagarange ...
3
votes
5answers
102 views

Analyze if this series converges: $\sum_{n=0}^{\infty}\frac{n^{2}+1}{n!}$

Analyze if this series converges: $\sum_{n=0}^{\infty}\frac{n^{2}+1}{n!}$ I have used ratio test: $\lim_{n\rightarrow \infty}\left |\frac{a_{n+1}}{a_{n}} \right |< 1$ $\Rightarrow$ $\lim_{n\...
3
votes
2answers
94 views

Very strange - what's the limit of $\lim_{x \rightarrow 0}\frac{sin(x)+cos(x)}{x}$?

What's the limit of: $\lim_{x \rightarrow 0}\frac{sin(x)+cos(x)}{x}$ ? $\lim_{x \rightarrow 0} \left (sin(x) + cos(x) \right) = sin(0)+cos(0) = 1 $ $\lim_{x \rightarrow 0} x = 0$ $\Rightarrow \frac{...
2
votes
0answers
20 views

Proving that $A \mapsto \sup\{ \mu E \mid A \supset E \in \Sigma, \mu E < \infty\}$ is an inner measure

Let $(X,\Sigma, \mu)$ be a measure space and define $m: 2^X \to [0,\infty]$ by $m A = \sup\{ \mu E \mid A \supset E \in \Sigma, \mu E < \infty\}$. Show that $m$ is an inner measure. There are $4$ ...
4
votes
2answers
30 views

Show that the following equation has got exactly one solution for each $C>0$

Show that the equation $$C=\left ( 1+x+\frac{1}{2}x^{2} \right)*e^{-x}$$ has got exactly one solution for each $C>0$. Alright so I did it like that but not sure if it's correct: $0<\left ...
4
votes
1answer
37 views

Definition second differential of a vector field

Let $f: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ be a smooth function. Then we know that its differential $df: \mathbb{R}^2 \rightarrow Hom(\mathbb{R}^2,\mathbb{R}^2)$ maps vectors to matrices/linear ...
6
votes
1answer
41 views

Limit of sequence $\lim_{n\to\infty}\frac{1+(\sqrt{n}+1)^{3}+2\sqrt{n}}{n+\sin(n)}$

This is no homework. It's another task of a sample exam and I'd like to know how to solve it. Find the limit of $$\lim_{n\to \infty}\frac{1+(\sqrt{n}+1)^{3}+2\sqrt{n}}{n+\sin(n)}$$ Both ...
5
votes
4answers
329 views

Is it possible / allowed to use L'Hôpitals rule for products?

In our readings, we had L'Hôpitals rule and defined it like that: $\lim_{x\rightarrow x_{0}}\frac{f'(x)}{g'(x)}$ Because we had it in our readings, we are allowed to use this to find limit of ...
6
votes
6answers
114 views

Proving that ${x +y+n- 1 \choose n}= \sum_{k=0}^n{x+n-k-1 \choose n-k}{y+k-1 \choose k} $

How can I prove that $${x +y+n- 1 \choose n}= \sum_{k=0}^n{x+n-k-1 \choose n-k}{y+k-1 \choose k} $$ I tried the following: We use the falling factorial power: $$y^{\underline k}=\underbrace{y(y-1)(...
1
vote
1answer
32 views

If for every $x\in\mathbb{R}^{3}$ the rank of the derivative $Df(x)$ is 2, prove that the image of $f$ is an open set.

I don't see how to solve the following problem, any suggestions? Let $f:\mathbb{R}^{3}\to \mathbb{R}^{2}$ such that $f\in C^{1}$. If for every $x\in\mathbb{R}^{3}$ the rank of the derivative $Df(...
3
votes
1answer
46 views

$\text{ Proving }\; A \subseteq \Bbb R \text{ A is bounded above} \Rightarrow A^c \text{ is not?} $

Prove: Let $A \subseteq \Bbb R$. Prove that if $A$ is bounded above, then $A^c$, the complement of $A$ is not bounded above. $ A^c = $ those element of the universe that are not in A. $ \Bbb R =$ ...