0
votes
0answers
25 views

A sequence $a_n$ converges to $+\infty$ iff it has $+\infty$ as it's only partial limit

In one of my books there was an exercise to prove that: A sequence $a_n$ converges to $+\infty$ iff it has $+\infty$ as it's only partial limit (I know a sequence doesn't really converge to ...
1
vote
3answers
58 views

Sequential compactness in $\mathbb{R}$

Well known result: Suppose $f:\mathbb{R}\to \mathbb{R}$ is continuous and let $K$ be a compact set. Then, $f(K)$ is compact. I can prove this using the definition of compactness (finding a ...
4
votes
0answers
55 views

If $f_n(t):=f(t^n)$ converges uniformly to continuous function then $f$ constant

Let $f \colon [0,1] \to \mathbb R$ be continuous and let $f_n$ be defined by $$ f_n(x):=f(x^n), \qquad x \in [0,1], \,\, n \in \mathbb N. $$ Suppose there exists a continuous function $g$ on ...
2
votes
2answers
90 views

A problem on nested radicals

Find the value of $x$ for all $a>b^2$ if: $$\large x=\sqrt{a-b\sqrt{a+b\sqrt{a-b{\sqrt{a+b.......}}}}}$$ My attempt $$\large x=\sqrt{a-b\sqrt{(a+b)x}}$$ $$\large x^4=(a-b)^2(a+b)x$$ $$\large ...
1
vote
1answer
24 views

General convergence of Sums

This is to be proven or disproven: Be $(a_n)_{n\in\mathbb{N}}$ a real sequence with $a_n \geq 0$ $ \forall n \in\mathbb{N}$. Then, if $\sum_{k=0}^{n} a_k$ converges for n$\to \infty$ also ...
2
votes
1answer
47 views

Answer check on two series

I want to determine if these two are absolutely convergent, conditionally convergent or simply divergent. 1) $$\sum_{n=2}^\infty \left(\frac xn - \frac x{n-1}\right)$$ $$= \frac x2 - \frac x1 + ...
4
votes
5answers
117 views

Showing a recursion sequence isn't bounded $a_{n+1}=a_n+\frac 1 {a_n}$

Show the sequence isn't bounded: $a_1=1$, $a_{n+1}=a_n+\frac 1 {a_n}$. Proof by contradiction: Let $M>0$ such that $\forall n: |a_n|< M$. Let $\epsilon >0 $ and for some $n=N, ...
5
votes
3answers
92 views

Proving $\limsup\frac 1 {a_n}=\frac 1 {\liminf a_n}$ and $\limsup a_n\cdot \limsup \frac 1 {a_n} \ge 1$

Let $a_n$ be a sequence such that $\forall n\in \mathbb n: 0<a\le a_n\le b <\infty.$ Prove: $\displaystyle\limsup_{n\to\infty}\frac 1 {a_n}=\frac 1 ...
4
votes
1answer
59 views

For what $\alpha$ does the series converge: $\sum^\infty_{n=2}\frac {1}{n^\alpha\log_2(n)}$

Let $\alpha\ge 0$ check for what $\alpha$ does the series converge: $$\displaystyle\sum^\infty_{n=2}\dfrac {1}{n^\alpha\log_2(n)}$$ I tried the condensation test and get: ...
2
votes
1answer
45 views

$a_{n+1}=a_n-a^2_n$ show the recursion sequence is convergent and find its limit

Let $a_1=\frac 2 3 , \ a_{n+1}=a_n-a^2_n$ for $n\ge 1$. Show the sequence is convergent and find its limit. In order to show convergence, I need to show that it's monotone and bounded. ...
1
vote
2answers
36 views

Proof Check: Every Cauchy Sequence is Bounded

Sorry if I keep asking for proof checks. I'll try to keep it to a minimum after this. I know this has a well-known proof. I understand that proof as well but I thought I'd do a proof that made sense ...
1
vote
1answer
33 views

How to show that $\sum_{n=1}^\infty (-1)^n\frac {x^2+n}{n^2}$ is uniformly convergent?

Show that $\sum_{n=1}^\infty (-1)^n\frac {x^2+n}{n^2}$ is uniformly convergent on arbitrary interval. I wanted to use M test for arbitrary [a,b] $|(-1)^n\frac ...
2
votes
2answers
57 views

Proving $a_n=\frac 1 2 \max\{a_{n-k},a_{n-k+1},…,a_{n-1}\}+1$ is monotone and finding its limit

Let $(a_n)^{\infty}_{n=1}$ defined like so: let $2\le k\in \mathbb N$ and $a_1,a_2...a_k\in \mathbb R$, $a_j\le 2, \ j=1,2,...,k$. Let $\forall n\ge k+1 : a_n=\frac 1 2 ...
1
vote
1answer
46 views

Prove that any unbounded sequence has a subsequence that diverges to $∞$.

To prove that any unbounded sequence has a subsequence that diverges to ∞, is it enough to say that you can take a subsequence $(a_{m(k)})$ where $m(k)=k$, as you know that this diverges to infinity, ...
0
votes
2answers
37 views

I know the limit of a subsequence exists, but how do I find it?

I know intuitively that for: $(a_n) = (1,1,2,1,2,3,1,2,3,4, ...)$ for any $m$ in the positive natural number there is a subsequence (m,m,m,..) that tends to m as n tends to infinity. I believe I ...
0
votes
1answer
40 views

Less than or equal summations

Hi,I want to prove above unequal that consist of two summation both of this sides.It a formula in Computer Network to control Congestion.The way to prove it is not important, but because I weak in ...
3
votes
0answers
61 views

Proof of Abel's theorem

I tried to prove: If $g(x) = \sum_{n=0}^\infty a_n x^n$ is a power series that converges at $x= R > 0$ then it converges uniformly on $[0,R]$. Please can you check my proof? Let $\varepsilon ...
1
vote
2answers
23 views

Series, limits and convergence.

Theorem $\,\bf3.3.1.\;$ If the series $$\sum_{n=1}^\infty a_n$$ is convergent then $\lim\limits_{n\to\infty}a_n=0$. Proof. Let $s_n=\sum_{k=1}^n a_k.$ Then by the definition the limit $\lim_n ...
1
vote
0answers
39 views

Is any of this true about infinite series of functions?

Let $f_n^+(x)$ be a sequence of non-negative functions $f_n^+: X \to \Bbb{R}_{\geq 0}$, such that each $f_n^+$ has countably many zeros. Then if $f(x) = \sum f_n^+(x)$ converges point-wise, the ...
2
votes
0answers
34 views

Find maximal possible sum of a tricky series

for $a \in R$, $n \in N$ let $a_n$ closest distance between $a$ and $\frac m {2^n}$, where $m \in Z$. Find maximal possible sum of a series: $\sum_{n=0}^\infty a_n$ I came up with solution for the ...
0
votes
1answer
36 views

Non-increasing Monotone Sequence Convergence Proof

My goal is to prove the monotone convergence of a non-increasing sequence of real numbers. There are some steps in the proof that I'm not sure about. The question: If $S$ is a non-increasing sequence ...
3
votes
1answer
35 views

Proof that if $\lim s_n=0$ then $\lim\sqrt{s_n}=0$

Here's the question: Suppose $(s_n)$ is a sequence of non-negative real numbers, and $\lim( s_n) = 0$. Prove that $\lim(\sqrt{s_n})=0$. Here's my proof. Can someone please verify it or offer ...
0
votes
1answer
39 views

If a sequence of functions converges uniformly, then its limit is bounded

Let $\{f_n\}_{n=1}^{\infty}$ be a sequence of functions defined on $[a,b]$ Assume that each $f_n$ is bounded, so $|f_n| \le M_n$ for all $x \in [a,b]$. If $\{f_n\}_{n=1}^{\infty}$ converges uniformly ...
3
votes
0answers
51 views

Proving $\sum^\infty_{n=1}a_n$ converges absolutely iff each each sub series converges

We have a series $\displaystyle\sum^\infty_{n=1}a_n$ and a sub series $\displaystyle\sum^\infty_{k=1}a_{n_k}$ where $n_k\in\mathbb N$. Prove that $\displaystyle\sum^\infty_{n=1}a_n$ converges ...
2
votes
1answer
48 views

Prove shuffled sequence $\{x_i, y_i\}$ converges $\iff \lim x_n = \lim y_n$ (Abbott p 49 q2.3.5)

Let $(x_{n})$ and $(y_{n})$ be given. Define $(z_{n})$ to be the shuffled sequence $(x_{1}.y_{1},\ x_{2},\ y_{2},\ x_{3},\ldots,x_{n}, y_{n},\ldots)$ . Prove that $(z_{n})$ is conv ergent $\iff ...
4
votes
1answer
57 views

Proving for all polynomials $p(x)=p_0+p_1x+…+p_dx^d$, $ \ \sum^\infty_{n=1}p(a_n)$ converges iff $p_0=0$

The series $\displaystyle\sum^\infty_{n=1}a_n$ converges absolutely. Prove that for every polynomial $p(x)=p_0+p_1x+...+p_dx^d$, $\ \displaystyle\sum^\infty_{n=1}p(a_n)$ converges iff ...
0
votes
2answers
88 views

Showing $a_n=\sin(n)$ does not converge

Show that $a_n=\sin(n)$ does not converge My idea: Take two subsequences: $a_{n_k}=\sin(\frac {\pi k} 2)$ , $a_{n_l}=\sin(\frac {2\pi l} 3)$ So: $\forall n$ : $\lim_{n\to\infty} a_{n_k}=1$, ...
0
votes
2answers
31 views

Help with proof by induction?

Prove $\frac{2(n-c)}{n+1} < 2$ where c is any natural So we assume $\frac{2(n-c)}{n+1} < 2$ is true, and so far I have $\frac{2(n+1-c)}{n+2} = \frac{2n-2c+2}{n+2} = \frac{2(n-c)}{n+2} + ...
1
vote
0answers
42 views

Convergence of geometric shapes

Let $c$ be a closed geometric shape. Let $P$ and $Q$ be two points in $c$. Let $S$ be the family of all closed squares. Let $s_1, s_2,...$ be a sequence of shapes from family $S$ such that for every ...
1
vote
2answers
56 views

l'Hopitals rule - is my working correct?

Is anyone able to help me with this question on l'Hopital's rule? Use l'Hopital's rule to find the limit of the sequence $\{a_n\}_{n=1}^\infty$ with $n$-th term $\displaystyle a_n = ...
4
votes
1answer
67 views

Show that $f(z)=\sum_{n= 0}^{+\infty}a_n z^n$ is a polynomial

Let $f(z)=\sum_{n= 0}^{+\infty}a_n z^n$, the radius of convergence $\ge 1$. For all $n,\quad a_n\in \mathbb{Z}$ and $f$ is bounded the open unit disk. Show that $f$ is a polynomial. My ...
1
vote
2answers
119 views

Proof Verification - Every sequence in $\Bbb R$ contains a monotone sub-sequence

Came across the following exercise in Bartle's Elements of Real Analysis. This is the solution I came up with. Would be grateful if someone could verify it for me and maybe suggest better/alternate ...
2
votes
1answer
109 views

Proof Verification: Show sequence is bounded and find limit: $x_1 \gt 1$ and $x_{n + 1} = 2 - \frac{1}{x_n}$

Came across the following exercise in Bartle's Elements of Real Analysis and am a little unsure about my solution. Would be extremely grateful if someone could verify it for me. Let $x_1 \in ...
0
votes
1answer
235 views

Waring's Inequality Solution

$$ \text{Waring's problem asks, "Is }\left\lfloor \left(\frac{3}{2}\right)^n\right\rfloor =\left\lfloor \frac{3^n-1}{2^n-1}\right\rfloor\text{ always true?"} $$ We craft an inequality, with $m,n ...
1
vote
2answers
43 views

$\pi \cot (\pi z) = \frac{1}{z} + \sum_{n \in \mathbb{Z},\ n \neq 0} \frac{1}{z-n}+ \frac{1}{n}$

I'm reading the proof that $$\pi \cot (\pi z) = \frac{1}{z} + \sum_{n \in \mathbb{Z},\ n \neq 0} \frac{1}{z-n}+ \frac{1}{n}$$ There is a function $$h(z) =\pi \cot (\pi z) -[ \frac{1}{z} + \sum_{n ...
5
votes
1answer
254 views

Tricks. If $\{x_n\}$ converges, then Cesaro Mean converges (S.A. pp 50 2.3.11)

Show if $\{x_n\}$ is a convergent sequence, then the sequences given by the averages $\{\dfrac{x_1 + x_2 + ... + x_n}{n}\}$ converges to the same limit. (Not a duplicate) Let $\epsilon>0$ be ...
1
vote
1answer
55 views

A rearrangement of an absolutely convergent complex series is also absolutely convergent

I just completed the following proof. Is it valid? Let $\sum_{k=1}^{\infty} a_k$ be an arbitrary convergent series that also converges absolutely. Then $\sum_{k=1}^{\infty} a_k \in \mathbb{C}$ and ...
1
vote
2answers
70 views

Help with a general solution for $\int \tan^a \theta$

Working out the integrals of $\tan^6 \theta$ I saw a pattern. I would like to put it in series representation. The pattern is as follows: $$\int \tan^a d\theta= \int\tan^{a-2}\theta(\sec^2\theta-1)\ ...
4
votes
1answer
115 views

Big Oh and Taylor polynomials.

Prove the following statement. Suppose $f$ has $n+1$ continuously differentiable derivatives on $[a,b]$, let $c \in (a,b)$. We define $P_n(x) = \sum_{k = 0}^{n} \frac{f^{k} (c) w^k}{k!}$. Then ...
1
vote
0answers
38 views

Proving that a convergent infinite series implies convergent part of that series and vice versa

I am asked to prove the following theorem: $\sum_{n = 1}^\infty a_n$ converges if and only if $\sum_{n=N}^\infty$ converges. I am not sure if I have done this correctly: Suppose that ...
3
votes
1answer
152 views

If $f$ is continuous on $[0,1]$ and if $\int\limits_{0}^{1} f(x) x^n dx = 0, (n=0,1,2,…)$, prove that $f(x)=0$ on $[0,1]$.

If $f$ is continuous on $[0,1]$ and if $\int\limits_{0}^{1} f(x) x^n dx = 0, (n=0,1,2,...)$, prove that $f(x)=0$ on $[0,1]$. This is what I have, how does it look? Proof: Let $P(x)$ be any ...
1
vote
0answers
50 views

How to prove: If $a \to -\infty $ and $b$ is bounded from below by a constant $k\in\Bbb R^{>0}$, then the $a\cdot b\to -\infty$

I must proof the following, with $a: \Bbb{N} \to \Bbb{R}$ and $b: \Bbb{N} \to \Bbb{R}$ If $a \to -\infty\ (n\to\infty)$ and $b$ is bounded from below by a constant $k\in\mathbb R^{>0}$, then the ...
0
votes
1answer
26 views

Proof-checking: $a \to +\infty \wedge \exists k \in \Bbb{R}(\forall r \in \Bbb{N}(k \leq b(r))) \Rightarrow (a+ b) \to+\infty$

let be $a: \Bbb{N} \to \Bbb{R}$, and $b: \Bbb{N} \to \Bbb{R}$, I must proof the following: "$a \longrightarrow +\infty \wedge \exists k \in \Bbb{R}(\forall r \in \Bbb{N}(k \leq b(r))) \Rightarrow ...
0
votes
1answer
60 views

Proof that a sequence has a convergent subsequence

I have a bounded sequence $a(n)$. We consider the set of all the values of $a(n)$ and let $M$ be the supremum of this set (without being one of its elements). Now we want to show that there is a ...
1
vote
1answer
84 views

Proving summation identities [duplicate]

How would one go about proving the following identities? $$\sum_{i=1}^n \sum_{i\neq j}^n \frac{z_i}{z_i-z_j} = \frac{n(n-1)}{2}$$ $$\sum_{i=1}^n \sum_{i\neq j}^n \frac{z_i^2}{z_i-z_j} = ...
2
votes
3answers
140 views

Theorem 9.7 - Uniform convergence and continuity of the limit function - Math Analysis 2nd ed - Apostol

Theorem 9.7 Assume that $\sum f_n(x)= f(x)$ (uniformly on $S$). If each $f_n$ is continuous at a point $x_0$ of $S$, then $f$ is also continuous at $x_0$. Proof. I define $s_n(x) = \sum_{k=1}^n ...
2
votes
1answer
107 views

Theorem 9.5 Cauchy Condition for uniform convergence of series - Math Analysis 2nd ed - Apostol

Theorem 9.5 (Cauchy condition for uniform convergence of series) The infinite series $\sum f_n(x)$ converges uniformly on $S$ if, and only if, for every $\epsilon>0$ there is an $N$ such that ...
5
votes
3answers
206 views

A tricky infinite sum— solution found numerically, need proof

Consider an infinite sum of the following form: $X Y^{\alpha} + X^2 Y^{\alpha + \alpha^2} +X^3 Y^{\alpha + \alpha^2 + \alpha^3} + ...$ ...which can be expressed more succinctly as: $\sum\limits_{j ...
2
votes
1answer
73 views

Recurrence relations - simple questions, please verify my answers.

I'm posting this question because this is new material for me and I am unsure of my answers and have no one to consult with. I solved the first three and would appreciate feedback. I need help solving ...
2
votes
0answers
184 views

Fubini's Theorem for Infinite series

In the book what I've read, there is one point where the author suggest to begin the proof of the Fubini's Theorem for infinite sum in the case when is non-negative after this try to generalize. But ...