0
votes
0answers
10 views

If $\lim_{n \to ∞} z_n=L_1$ then $\lim_{n \to ∞} \overline{z_n}=\overline{L_1}$ for $z_n \in \mathbb{C}$

I tried proving that if $\lim_{n \to ∞} z_n=L_1$ then $\lim_{n \to ∞} \overline{z_n}=\overline{L_1}$ for $z_n \in \mathbb{C}$. This is my attemt. Let $\epsilon>0$. Then there exists $N\in ...
0
votes
3answers
21 views

Bounded Sequences and Extrapolation of Convergence From Related Sequences

I'm considering some sequence $S_n$ which is bounded, and I want to prove that $S_n/n$ is convergent. I'm thinking that I could simply take $lim_{n \to \infty} S_n/n$ and simplify this to $(lim_{n \to ...
3
votes
2answers
40 views

Limit of a sequence proof by contradiction

Suppose I have a monotonically decreasing sequence $a_{n}$ such that $a_{n}$ is positive for all $n \in \mathbb{N}^{+}$ and that $$\lim_{n\rightarrow \infty} \frac{a_{n+1}}{a_{n}} = 0.$$ It seems to ...
2
votes
1answer
41 views

If $a_n=n^\frac 1n-1, n \in \mathbb N$ prove that $0 \le a_n \le \sqrt {2/n}$?

If $a_n=n^{\frac{1}{n}}-1$, $n\in\mathbb{N}$, prove that $0\le a_n\le\sqrt{\frac{2}{n}}$. I tried with induction and signs, got nowhere. Any help is appreciated.
6
votes
6answers
114 views

Show that $\lim_{n\to\infty}\frac{a^n}{n!}=0$ and that $\sqrt[n]{n!}$ diverges.

Let $a\in\mathbb{R}$. Show that $$ \lim_{n\to\infty}\frac{a^n}{n!}=0. $$ Then use this result to prove that $(b_n)_{n\in\mathbb{N}}$ with $$ b_n:=\sqrt[n]{n!} $$ diverges. ...
0
votes
0answers
19 views

Proving a property of the largest limit point

Let $(a_n)_{n\in\mathbb{N}}$ be a bounded sequence. By Bolzano-Weierstraß this sequence does have a limit point. Let $\bar{a}$ denote the largest limit point of the sequence. Show that among ...
2
votes
1answer
23 views

Proving $\{x_n\}$ converges to $a$ when $|x_n-a|\le Cb_n$ for large $n$ and $C$ is a positive constant.

If $\{b_n\}$ is a sequence of nonnegative numbers that converges to $0$, and $\{x_n\}$ is a real sequence that satisfies $|x_n-a|\le Cb_n$ for large $n$, where $C$ is a fixed positive constant, prove ...
0
votes
0answers
51 views

Prove $x_n$ converges to $a$ iff every subsequence of $x_n$ also converges to $a$.

Prove $x_n$ converges to $a$ iff every subsequence of $x_n$ also converges to $a$. Suppose that $\{x_n\}$ is a sequence in $\mathbb R$. Definitions available: (1) A sequence of real numbers ...
1
vote
1answer
52 views

Attempt to prove that every real number is a limit of a sequence of rational numbers

Prove that given a real number $x$, there exists a rational sequence $r_n$ such that $r_n \to x$ as $n$ grows. Proof: Suppose $x$ is a real number. Then we know by definition, there exists a ...
2
votes
1answer
43 views

Proving $\sum_{k=1}^{\infty}\frac{\sin kx}{x}=\frac{\pi-x}{2}$ for $0\le x\le 2\pi$

Refer to this OP: Sign of a series, we have the following equation \begin{equation} \sum_{k=1}^{\infty}\frac{\sin kx}{k}=\frac{\pi-x}{2} \end{equation} defined for $0\le x\le 2\pi$. Here is ...
5
votes
2answers
46 views

$(u_{2n})$,$(u_{2n+1})$,$(u_{3n+1})$ converge $\underset{???}{\Rightarrow}(u_n)$ converges

Let $(u_n)_{n_\in\mathbb{N}}\in\mathbb{C}^\mathbb{N}$. We know that $(u_{2n})$, $(u_{2n+1})$ and $(u_{3n+1})$ converge. The question is to know whether $(u_n)_{n\in\mathbb{N}}$ converges. /!\ I ...
0
votes
2answers
18 views

Convergence of a sequence in absolut value.

I need to prove this: If $a_{n}$ converges to $A$, then $|a_{n}|$ converges to $|A|$. And I have this: $a_{n} \rightarrow A$ then, given $\epsilon>0$ there exists $N \in J$ such that ...
0
votes
0answers
26 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
59 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
97 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
50 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
120 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
114 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
60 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
50 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
40 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
34 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
61 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
58 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
38 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
41 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
66 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
24 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
42 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
36 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
43 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
52 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 ...
3
votes
1answer
55 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
61 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
98 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
47 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
68 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
130 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
116 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
237 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
44 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
264 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
56 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)\ ...