For questions concerning a specific proof, asking for verification, identifying errors, suggestions for improvement, etc.

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Convergence with Ratio Test (β, α)

Decide whether the series $$\sum_{n = 1}^\infty \frac{α^n}{n^β}$$ converges absolutely, conditionally, or diverges. (May depend on α and β, consider all cases α,β in R ). Started with Ratio Test to ...
3
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1answer
31 views

$\sum_{n=0}^\infty z^n = \prod_{m=0}^\infty \left(1+z^{2^m}\right)$

When reading Iwaniec and Kowalski's Analytic Number Theory, I came across the following "identity" on page 11 (the Amazon link has a free book preview which includes page 11): $$\sum_{n=0}^\infty z^n ...
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2answers
16 views

Let $\text{A}$ be a nonsingular $\textit{n}\times\textit{n}$ matrix, and let $\textit{B}$ be a basis for $\mathbb{R}^n$

Show that $ B_1 = \{\textbf{Av}| \textbf{v} \in B\} $ is also a basis for $\mathbb{R}^n.$ I apologize for my informality, but I would really like some feedback as to whether I am using the correct ...
1
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1answer
38 views

Prove there exists a $c$ such that $-f'(c)=\frac{f(c)}c$

Let $f: \Bbb{R} \longrightarrow \Bbb{R}$ be a continuous function on $[0,2]$ and differentiable on $(0,2)$. $f(2)=0$. Prove there exists a $c \in (0,2)$ such that $-f'(c)=\frac{f(c)}c$. What I did: ...
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3answers
54 views

Show that $\sin \dfrac{n \pi}{4}$ is divergent.

Show that $\sin \dfrac{n \pi}{4}$ is divergent. My attempt: Consider the subsequences $x_{4n}=\sin (n \pi)$, which converges to $0$, and $x_{8n+2}=\sin \dfrac{2(4n+1) \pi}{4}$, which converges ...
0
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0answers
8 views

Sign Of Permutation That Is Written As C Different Cycles

prove: if $\sigma\in S_n$ is a factorization of $c$ disjoint cycles so $Sgn(\sigma)=(1)^{n-c}$ We know the one cycle sign is $(-1)^{l-1}$ so $c$ of them is $(-1)^{l-1}\cdot ...
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2answers
13 views

Suppose that $A$ and $B$ are subsets of $X$ such that $A \subseteq B$ then $Int(A) \subseteq Int(B)$.

Let $(X, \mathfrak T)$ be a topological space and suppose that $A$ and $B$ are subsets of $X$ such that $A \subseteq B$ then $Int(A) \subseteq Int(B)$. I know this a true statement so now I need to ...
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0answers
11 views

Apply the Euclidean algorithm to $2$ and $1 − 3i$ in the integers of $\mathbb Q[\sqrt {-1}]$

Question: I have to apply the Euclidean algorithm to $2$ and $1 − 3i$ in the integers of $\mathbb Q[\sqrt {-1}]$ My Solution: Since $N(2) = 4$ and $N(1–3i) = 10,$ we must start by dividing $1–3i$ ...
0
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0answers
27 views

Proving if $f(x)$ is an integrable function on $[a,b]$ then $g(x)=f(x-c)$ is integrable on $[a+c,b+c]$

Prove that if $f(x)$ is an integrable function on $[a,b]$ then $g(x)=f(x-c)$ is integrable on $[a+c,b+c]$. My attempt: Since $f$ is integrable then there's a sequence of partitions ...
0
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1answer
9 views

Distance between sets in $\mathbb{R}^p$

Let $A$ and $B$ be closed, bounded, disjunct subsets of $\mathbb{R}^p$ Now, this is not a metric, but define $\delta$ like this: $$ \delta = \inf V, $$ where $$ V = \{ \| a-b \| \mid a \in A \text{ ...
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0answers
6 views

Proof regarding representation of numbers

Prove directly that two different representations to the base k represent different integers. I cannot use the basis representation theorem, only the fact that $a_sk^s + a_{s-1}k^{s-1} + ... + a_0 = ...
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2answers
32 views

If $(a_n)$ is an increasing sequence, $(b_n)$ is a decreasing sequence, $a_n \leq b_n \forall n \in \mathbb N$. Prove $\lim a_n \leq \lim b_n$

If $(a_n)$ is an increasing sequence and $(b_n)$ is a decreasing sequence, with $a_n \leq b_n \forall n \in \mathbb N$. Prove that $\lim a_n \leq \lim b_n$ This is the closest thing I found on this ...
1
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2answers
28 views

Proving that a nonzero integer n has a unique representation

This is the first proof I've written. Can anyone give me advice? I don't know if its valid, or if there are ways to improve / other ways to do it: Prove that each nonzero integer may be uniquely ...
1
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1answer
45 views

Is this definition valid?

I am working on this problem: "Suppose $f:A\times A\rightarrow A$. A set $C \subseteq A$ is closed under $f$ if $\forall (x,y) \in C \times C(f(x,y) \in C)$. Now suppose $B \subseteq C $. The closure ...
0
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1answer
49 views

How many numbers smaller than one million, their sum of digits is at least 20?

How many numbers smaller than one million, their sum of digits is at least 20? My attempt: Since I don't know how to handle the "at least" part, I'll be using a complement: The general case is ...
0
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1answer
21 views

Check my logical argument for this proof.

if x is a real number $x \not =\ 1 $, then there exists y which is also a real number $ ((y+1) \div ( y-2) ) = x .$ Prove it's converse also. Logical Argument: given: $x \not = 1$ Goal: $ ...
0
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1answer
17 views

Let $(X,\mathfrak T)$ be a topological space and suppose that A and B are subsets of X such that $A \subseteq B$ then $Cl(A) \subseteq Cl(B)$.

Let $(X,\mathfrak T)$ be a topological space and suppose that A and B are subsets of X such that $A \subseteq B$ then $Cl(A) \subseteq Cl(B)$. My definition of closure is "Let $(X, \mathfrak T)$ be ...
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0answers
22 views

Let $X=\mathbb Z^+$. Topologize $X$ with the finite complement topology. Let $A=\{3,5,6,7\}$. Find $\overline A$, int$(A)$, and bdry$(A)$.

Let $X=\mathbb Z^+$. Topologize $X$ with the finite complement topology. Let $A=\{3,5,6,7\}$. Find closure of $A$ $(\overline A)$, interior of $A$ (int$(A)$), and boundary of $A$ (bdry$(A)$). $A$ ...
2
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1answer
28 views

Set theory (containing Power Set) Need Help in a proof

I am confirming whether my proof is correct or not and need help. If $ A \subseteq 2^A , $ then $ 2^A \subseteq 2^{2^A} $ Proof: Given: $ \forall x ($ $ x\in A \rightarrow \exists S $ where $ ...
1
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1answer
30 views

If $(x_n)$ is a convergent sequence, for any $\epsilon>0, \exists M$ such that $|x_n-y_n|<\epsilon$ for all $n \geq M$. Is $(y_n)$ convergent?

If $(x_n)$ is a convergent sequence and $(y_n)$ is such that for any $\epsilon>0, \exists M$ such that $|x_n-y_n|<\epsilon, \forall n \geq M$. Is $(y_n)$ convergent? My attempt: Let lim ...
0
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1answer
17 views

Give example of convergent and divergent sequences of positive numbers such that lim $x_n^{\frac1n}=1$

So basically I have to give examples where the root test doesn't work, right? For the convergent sequence, I'm taking $x_n=(1,1,1,\cdots)$, then lim $x_n^{\frac1n}=1$ Is there any non-constant ...
2
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1answer
24 views

Simple true/false statements about function composition

Given the functions $f,g,h$ from $\mathbb{R}$ to $\mathbb{R}$ I have to determine whether the following statements are true: "If $f \circ g$ is strictly increasing and $f$ is injective then $g$ is ...
2
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2answers
29 views

Give an example to show that convergence of $|x_n|$ does not imply the convergence of $x_n$

I'm taking $x_n=(-1)^n=(-1,1,-1,1,\cdots)$, which is divergent, but $|x_n|=(1,1,1,1,1,\cdots)$ converges to $1$. Is this example correct?
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0answers
6 views

If a set A is bounded from above, then the set of upper bouds M has minimum

I hope the title is clear, because I am italian and I study calculus exclusively from italian books. I had to prove this proposition refusing to look the book (because if I read the proof, I'll ...
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1answer
12 views

Question on the argument proving primary decomposition theorem

Lang - Algebra p.150, Lemma 7.6 Let $E$ be a torsion module of exponent $p^r(r\geq 1)$ for some prime element $p$. Let $x_1\in E$ be an element of period $p^r$. Let $\bar E = E/(x_1)$. Let ...
0
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1answer
17 views

Under What Conditions Is $f:M\rightarrow \mathbb{C}$ Where M Is the Set of 2x2 Matrices a Function and Not a Function?

I came across a problem that I thought was interesting. I attempted to solve the problem below, and I would be grateful if someone would check my logic in what follows. Let the set M of all 2 by 2 ...
0
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0answers
15 views

In the field $GF(p^n)$, prove that for every positive divisor d of n, $x^{p^n }-x$ has an irreducible factor over $GF(p)$ of degree d

In the field $GF(p^n)$, prove that for every positive divisor d of n, $x^{p^n }-x$ has an irreducible factor over $GF(p)$ of degree d Let d be any divisor of n. Then $GF(p^d)$ is a subfield of ...
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2answers
44 views

If a group G is isomorphic to H, prove that Aut(G) is isomorphic to Aut(H)

If a group G is isomorphic to H, prove that Aut(G) is isomorphic to Aut(H) Properties of Isomorphisms acting on groups: Suppose that $\phi$ is an isomorphism from a group G onto a group H, then: ...
3
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2answers
31 views

How to prove this Dirac Delta Function property?

How to prove the equation below, using Dirac Delta function properties? $$ \delta(x^2-m^2)=\frac{1}{2|w|}(\delta(x-w)+\delta(x+w)) $$ where $$ w^2=|x|^2+m^2 $$ I tried to show it using $$ ...
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1answer
28 views

The Zeros Localization Theorem and the Extreme value Theorem

My intro to analysis book calls the following theorem The Zeros localization theorem: Theorem: Let $p(x)=x^n + a_{n-1}x^{n-1} +\dots+a_1 +a_0,\ x \in \mathbb R$ , be a polynomial. Then all the zeros ...
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0answers
10 views

Proving that equilateral triangle has equal medians. [on hold]

How to prove that equilateral traingle has equal medians? Mathematical method. Thank you. :D
1
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1answer
24 views

Describe explicitly the linear transformation T from $F^2$ to $F^2$ such that $T\epsilon_1=(a,b),T\epsilon_2=(c,d)$

I have to describe explicitly the linear transformation T from $F^2$ to $F^2$ such that $T\epsilon_1=(a,b),T\epsilon_2=(c,d)$ My try: We know that $T\epsilon_1=(a,b),T\epsilon_2=(c,d)$ so let ...
2
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2answers
49 views

Proof verification: ab = gcd(a,b)lcm(a,b) without use of prime factorization

I am trying to proof $ab = gcd(a,b)lcm(a,b)$. The definition of lcm(a,b) is as follows: t is the lowest common multiple of a and b if it satisfies the following: i)a | t and b | t ii)If a | c and ...
3
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1answer
26 views

real analysis question on equicontinuity

Can anyone verify my proof of the following problem found in Rudin's Principles of Mathematical Analysis chapter 7 exercise 15. Suppose $f$ is a real continuous function on $\mathbb{R}$, ...
0
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0answers
19 views

Find the upper sum and lower sum for the following function with respect to the given partition

I have the following homework problem: Find the upper sum and lower sum for the following function with respect to the given partition: Let $s:[0,1]\rightarrow \Bbb R$ be defined by: ...
0
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1answer
21 views

Proof Check: Alternating Series Approximation Theorem

Problem: Let $S=\sum\limits_{n=1}^\infty a_n$ be an alternating series where $\vert a_{n+1} \vert < \vert a_n \vert$ and $\displaystyle\lim_{n \rightarrow \infty}a_n=0$. Let $S_n$ be the $n$th ...
5
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4answers
57 views

Different Law of Cosines using Sine instead: $c^2 = a^2 + b^2 - 2ab\sqrt{1-sin^2(\theta)}$

Playing around with Trig and the Law of Cosines (LoC), I came up with this formula given a triangle with sides $a$, $b$, $c$ where we are given $a$, $b$ and angle $\theta$ between them: $$c^2 = a^2 + ...
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0answers
29 views

Prove that for every $n \in \mathbb N$, $\int_{-\pi}^{\pi}f(x)\sin(2nx)dx=0$ if $f(x)$ is odd.

If $f:\mathbb R \to \mathbb R $ is odd continuous function such that $g(x):=f(x + \frac{\pi}{2})$ is even, prove that for every $n \in \mathbb N$, $\int_{-\pi}^{\pi}f(x)\sin(2nx)dx=0$. Since ...
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2answers
24 views

Should the order of $a^k$ be $h/k$ as opposed to $h/(h,k)$?

Previously shown: Let $m\in\mathbb{N}$ and $a\in\mathbb{Z}$ s.t. $(a,m)=1$. Then the order $d$ of $a$ modulo $m$ exists and $d\mid\phi(m)$. Moreover, whenever $a^k\equiv 1\pmod{m}$, one has $d\mid ...
0
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1answer
35 views

Is the following proof rigorous? How could I improve or correct it?

Proof that $\sqrt{2}$ is irrational using the unique prime factorization theorem. My proof: Assume for the purpose of contradiction that $\sqrt{2}$ is rational. By the unique prime fact. the., we ...
2
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1answer
20 views

Prove that $A\subseteq B$ if and only if $A^{C}\cup B=\mathscr{U}$

Prove that $A\subseteq B$ if and only if $A^{C}\cup B=\mathscr{U}$. I know we have to show that: if $A\subseteq B$ then $A^{C}\cup B=\mathscr{U}$ if $A^{C}\cup B=\mathscr{U}$ then $A\subseteq ...
1
vote
1answer
47 views

$(B_X,w)$ metrizable implies $X^\ast$ separable

Let X be a normed space and assume that $(B_X,w)$ is metrizable, i.e. the weak topology is metrizable. Show that $X^\ast$ is separable. My attempt: Let $d$ a equivalent metric on $B_X$. For fixed ...
3
votes
2answers
56 views

If $\lim\limits_{z \to \infty} p(z) = \infty$, then $p(z)$ is a constant

Claim: If $p$ is an entire function and $\lim\limits_{z \to \infty} p(z) = \infty$ and $p(z) \neq 0$ $\forall z \in \Bbb C$, then $p(z)$ is a constant. Proof: Define $f(z) = \frac{1}{p(z)}$ so ...
3
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3answers
111 views

Proof verification for proving $\forall n \ge 2, 1 + \frac1{2^2} + \frac1{3^2} + \cdots + \frac1{n^2} < 2 − \frac1n$ by induction

Prove by mathematical induction: $\forall n \ge 2, 1 + \frac1{2^2} + \frac1{3^2} + \cdots + \frac1{n^2} < 2 − \frac1n$ Basis Step: (We want to show, $P(2)$, which is 1 + ...
2
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1answer
22 views

A clarification of addition on elliptic curves over the complex numbers

I am trying to prove that the order of the two points $P_{\pm}=(0,\pm\sqrt{-g_3})$ is three on the elliptic curve $y^2=4x^3-g_3$, for $g_3 \not= 0$, defined over $\mathbb{P}^2_{\mathbb{C}}$. Here's an ...
2
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2answers
38 views

Set Theory Proof: Valid or not?

I'm trying to gain understanding of set proofs and I came across this one. I can't help but think the proof is too simple and that there is more to it. Problem: Prove or disprove for arbitrary sets ...
1
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1answer
38 views

Show that if $p$ is an odd prime, then the congruence $x^2\equiv1\pmod{p^{\alpha}}$ has only two solutions, $x\equiv1,x\equiv-1\pmod{p^{\alpha}}$.

Show that if $p$ is an odd prime, then the congruence $x^2 \equiv 1 \pmod{p^{\alpha}}$ has only two solutions, which are $x \equiv 1, x \equiv -1 \pmod{p^{\alpha}}$. Clearly $x \equiv 1, x \equiv ...
2
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2answers
42 views

Limit proof check, show $f$ is bounded in a neighborhood of its limit point

edit: as lem has pointed out, the case where x=c is not handled. Could someone suggest an idea? Prove that if a function $f : A \to \mathbb{R} $ has a limit $l \in \mathbb{R} $ at $c \in L(A)$, then ...
2
votes
3answers
95 views

Showing that $f_0 (x_1, \ldots, x_m) \mathrm tr A = \sum_{i=1}^n f_0(x_1, \ldots, Ax_i,\ldots, x_m)$

Question: Consider $f: (-\epsilon, \epsilon) \to \mathbb R^{m^2}$ a differentiable path of matrices $m \times m$ such that $f(0) = I_m$ and the function $g: I \to \mathbb R$ is defined by $$g(t) = ...
2
votes
6answers
80 views

Formally proving that if $x^2 + 1$ is even, then $x$ is odd.

Theorem: If $x^2 + 1$ is even, then $x$ is odd. I have to mention, that I am a beginner at this. So, sorry if it is very wrong. Suppose that $x^2+1$ is even, such that there exists an ...