For questions about recurrence relations, convergence tests, and identifying sequences. For questions on finite sums use the (summation) instead.

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25 views

Series and integrals for inequalities and approximations to $\pi$

Fundamentals Two beautiful expressions that relate $\pi$ to its convergents are Dalzell integral $$\frac{22}{7}-\pi=\int_0^1\frac{x^4(1-x)^4}{1+x^2}dx$$ (see Why do we need an integral to prove ...
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2answers
46 views

Sequence of Partial Sums is Convergent

How do I show that the series $\sum_{n=1}^{\infty} \frac1{(2n-1)^{n}} + \frac1{(2n)^{3 }} $ is convergent? I'm trying to use Comparison Test but I'm having a hard time looking for a convergent series ...
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1answer
41 views

Does the convergence of a series $\sum_{n=1}^{\infty} x_n$ imply the convergence of the alternating series $\sum_{n=1}^{\infty} (-1)^{n} x_n $? [on hold]

True or False: If $\sum_{n=1}^{\infty} x_n$ is convergent, then $\sum_{n=1}^{\infty} (-1)^{n} x_n $ is convergent.
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1answer
28 views

Show that $ f(x)=\sum_{n=1}^{\infty} 2^{-n} f_n(x)$ defines a continuous function on $(0,\infty)$

Let $f_n$ be a sequence of continuous functions on $(0,\infty)$ with $|f_n(x)|\le n$ for every $ x>0$ and $n\ge1$, and such that $\lim_{x\to\infty} f_n(x) =0$ for each $n$.Show that $ ...
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1answer
18 views

Proving divergence of an alternating series with a non-standard function

I was given the following equation and told to prove either convergence or divergence. I am sure that it diverges, but I am unsure as to how I would go about proving that mathematically. $$ ...
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3answers
38 views

Finding the limit of recursive sequence

Assuming that the solution of $e^{-x}=x$ is $c\in (0,1)$ And give the following sequence $$a_{n+1}=e^{-a_n}$$ $$a_1=1$$ How can i prove that the sequence converge and that the limit is $$\lim_{n ...
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1answer
23 views

Proving that the sequence converges

I would like some help with the following problem. Thanks for any help in advance. Let $(x_n)$ and $(y_n)$ be convergent sequences of positive real numbers. Let $ x_n \xrightarrow[n \to \infty]{} x$ ...
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0answers
20 views

The critical value of an infinte sum over [0,1).

Consider $f(s)=\displaystyle\sum_{i=1}^\infty r_i^s$ where $s\in[0,\infty)$ and $0<r_i<1$. Under what conditions can we claim that there exists some $s$ such that $f(s)=1$. I know that some ...
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1answer
23 views

Prove that $\liminf_{n\rightarrow\infty}s_n\leqslant \liminf_{n\rightarrow\infty}\sigma_n$ for $\sigma_n=n^{-1}(s_1+…+s_n)$.

For my math class I have to prove that $\liminf_{n\rightarrow\infty}s_n\leqslant \liminf_{n\rightarrow\infty}\sigma_n$ and ...
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4answers
72 views

How to prove $ 1+1 + 2+ 2^2 +2^3 + \cdots + 2^{30}= 2^{31}$? [on hold]

This is true? $ 1+1 + 2+ 2^2 +2^3 + \cdots + 2^{30}= 2^{31}$, How to prove?
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0answers
25 views

Give an example to show that $f_n$ fails to converge to $f$ uniformly over $S$ if $S$ is not compact

Given the theorem: Suppose $S \subset \Bbb R^n$ is compact, and $P$ is an equicontinuous sequence of functions ($f_n$) over $S$ converging pointwise to a function $f$ at each $x \in S$, then $f_n$ ...
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0answers
27 views

Check the proof of $\Bbb R$ as set of subsequential limits

I want to prove that there is a sequence in $\Bbb R$ that has all of $\Bbb R$ as its set of subsequential limits. Could someone help me check my proof? If it's not correct, could someone give a proof? ...
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0answers
11 views

Prove that $\left\|f_n(x)-g_n(x)\right\|^2 = \|f_n(x)\|^2+ \|g_n(x) \|^2-2\operatorname{Re}\int_{\mathbb R} f_n(x)\overline{g_m(x)} dx$

Let $\{f_n(x)\}_{n\in\mathbb Z}$, $\{g_n(x)\}_{n\in\mathbb Z}$ be two sequence of square-integrable functions: $f_n, g_n\in L^2(\mathbb R)$. Prove that $$\left\|f_n(x)-g_n(x)\right\|^2_{L^2(\mathbb ...
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2answers
49 views

An example of a product of two distinct convergent series that is divergent

Do you have an example of a product of two distinct convergent series $\sum x_n$ and $\sum y_n$ $(y_n ≥0)$ that is divergent?
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2answers
38 views

Finding asymptotic relationship between: $\frac {\log n}{\log\log n} = (?) (\log (n-\log n))$

Given $f(n)=\frac {\log n}{\log\log n} , g(n)= (\log (n-\log n))$, what is the relationship between them $f(n)=K (g(n))$ where "K" could be $\Omega,\Theta,O$ I thought of taking a log to both ...
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0answers
19 views

$k$ be a positive integer ; $\{a_n\}$ be a sequence of positive integers such that $\gcd(m,n)=1 \implies a_m^k+a_n^k|m^k+n^k$ ; then $a_n=n$?

Let $k$ be a positive integer ; $\{a_n\}$ be a sequence of positive integers such that g.c.d.$(m,n)=1 \implies a_m^k+a_n^k|m^k+n^k$ ; then is it true that $a_n=n , \forall n \in \mathbb N$ ?
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1answer
23 views

Riemann zeta-function functional equation proof

I'm reading through Titchmarch's "The Theory of the Riemann Zeta-Function" and there's a part in the functional equation proof number 3 that I haven't figured out. He defines a function ...
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1answer
32 views

$ \sum_{n=1}^\infty \frac{x^4}{(1+x^4)^n} $ converges pointwise on $ R$, but not uniformly on $R$.

Prove Or Disprove, The series summation $$ \sum_{n=1}^\infty \frac{x^4}{(1+x^4)^n} $$ converges pointwise on $ R$, but not uniformly on $R$. I am struggling on this problem in real analysis ...
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0answers
28 views

approximation of a trigonometric sum

I have a trigonometric sum as below $$\sum_{r=0}^{N-1}\frac{\sin^2(\frac{\pi(Ne-e-r+n))}{N})}{\sin^2(\frac{\pi(r-n+e))}{N})}$$ and I want to show analytically that for small $e$ ($e<0.2$) and large ...
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4answers
52 views

To prove $\sum_{n=0}^\infty \binom{r}{x}\binom{N-r}{n-x}=\binom{N}{n}.$ [duplicate]

To prove $$\sum_{x=0}^n \binom{r}{x}\cdot \binom{N-r}{n-x}=\binom{N}{n}.$$ I tried comparing the coefficients of $(1+x)^{(n+k)} = (1+x)^n(1+x)^k$ but couldn't reach the answer.
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1answer
32 views

What is the $\lim _{n\to \infty \:}\left(\sum ^{n-1}_{j=1}\:\frac{3^{\frac{j}{2}}}{j!}\right)$?

We want to calculate $\lim _{n\to \infty \:}\left(\sum ^{n-1}_{j=1}\:\frac{3^{\frac{j}{2}}}{j!}\right)$. After converting the sum to $\sum ^{n-1}_{j=1}\:\frac{\sqrt 3 ^j}{j!}$ once can already ...
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1answer
33 views

Need help with a second-degree Taylor polynomial

It says to let T2(x) be the second degree polynomial for the functionf(x) = 6 + xe4x where a=0. I need to find T2(1). I thought it was just a taylor expansion and look at the second term, which I ...
0
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3answers
98 views

Show that $a_n = 1 + \frac{1}{2} + \frac{1}{3} +\dotsb+ \frac{1}{n}$ is not a Cauchy sequence [on hold]

Let $$ a_n = 1 + \frac{1}{2} + \frac{1}{3} + \dotsb + \frac{1}{n} \quad (n \in \mathbb{N}). $$ Show that $a_n$ is not a Cauchy sequence even though $$ \lim_{n \to \infty} a_{n+1} - a_n = 0 ...
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0answers
46 views

Convergence or divergence of infinte power towers of complex numbers $z^{z^{z^{z{…}}}}$

Let $s$ be any complex number, $t = e^s$ and $z = t^\frac{1}{t}$. Define the sequence $a_n$ by $a_0 = z $ and $a_{n+1} = z^{a_n} $ for $n > 0$, that is to say $a_n$ is the sequence $z$, $z^z$, ...
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1answer
121 views

A series to prove $\frac{22}{7}-\pi>0$

After T. Piezas answered Is there a series to show $22\pi^4>2143\,$? a natural question is Is there a series that proves $\frac{22}{7}-\pi>0$? One such series may be found combining ...
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1answer
25 views

Proof verification: Compact set has sup and inf

I was reading this post compact set always contains its supremum and infimum There was an answer reposted as follows: As $K$ is compact, we have that $K$ is bounded. So $\sup K$ and $\inf K$ ...
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1answer
23 views

Show that a sequence $\{s_n\}$ converges to $L$ if and only if the sequence $\{-s_n\}$ converges to $-L$.

I understand that both parts of this biconditional must be proven. If I assume that a sequence $\{s_n\}$ converges to $L$,then for every for every $ϵ>0$, there is some integer N where ...
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0answers
28 views

Root and Ratio Tests (Rudin)

In Rudin's presentation of the Ratio test, he implicitly assumes that $\{a_{n}\}$ is a sequence of nonzero (complex) numbers, for otherwise the ratio $|a_{n+1}/a_{n}|$ does not make sense for every ...
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1answer
16 views

Proving that if the sequence $\{s_n-L\}$ converges to zero, then a sequence $\{s_n\}$ converges to a limit $L$

I am having trouble proving this statement without using the limit rules. I know I start by assuming that the sequence $\{s_n-L\}$ converges to zero, therefore, for every number $ ϵ > 0 $, there ...
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1answer
122 views

How do I convert $1 - 1 + 1 - 1 + …$ to summation notation?

I can convert $1 + 2 + 3 + 4 + 5 + ... = -\frac {1}{12}$ to summation notation: $$\sum_{n = 1}^\infty n = -\frac {1}{12}$$ But, how can I convert the following series to summation notation: $$1 - 1 + ...
2
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2answers
35 views

Why is this the closed-form solution for this series? [duplicate]

I know this is simple, but I don't know very much at all about series, and I'm wondering how it's shown that: $$ 1 + 2 + 3 + \cdots + (n - 1) = \frac{n(n - 1)}{2} $$
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0answers
39 views

Show that the sequence of functions $f_n(x)=xe^{-nx}$ for $x\in(0,1)$ converges pointwise to $0$

Is it enough to calculate $\lim_{n \rightarrow \infty} xe^{-nx}$ or should we analyze it more carefully?
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1answer
35 views

Prove that $\sum_{n=1}^{\infty} \frac{[nx]}{n^2} $ is discontinuous at $x \in \mathbb Q$

$[x] := x - \lfloor x \rfloor$. I can prove that it is continuous at all irrational points using uniform convergence, but I don't know how to prove discontinuity in this case. I looked at this similar ...
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0answers
22 views

Extending real numbers with divergent sequences

Real numbers are defined through use of convergent sequences: each convergent sequence defines a real number. What if we postulate that every divergent sequence defines a "number" of some extended ...
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1answer
55 views

Prove that the series $\sum_{k=0}^\infty (-1)^k\ \left(\sqrt{k+1}-\sqrt{k}\right)$ converges but not absolutely.

I have to prove that the following series converges but not absolutely: $$\sum_{k=0}^\infty (-1)^k\ \left(\sqrt{k+1}-\sqrt{k}\right)$$ I have used the Leibniz test (alternating series test) to prove ...
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2answers
45 views

Question about proof: Uniform cauchy $\Rightarrow$ Uniform convergence

I have one quick question regarding the proof of a theorem contained in here : https://www.math.ucdavis.edu/~hunter/m125a/intro_analysis_ch5.pdf Theorem 5.13. A sequence $(f_n)$ of functions $f_n ...
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2answers
71 views

Prove $\lim_{n \to \infty} \frac{4n^3}{2n^2+1} \sin(\frac{\pi}{n}) = 2\pi$

For a beginning calculus student, prove $\lim_{n \to \infty} \frac{4n^3}{2n^2+1} \sin(\frac{\pi}{n}) = 2\pi$ I'm guessing this means something like Allowed: Pre-university maths, precalculus, ...
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2answers
96 views

New pattern question: $2, 7, 26, 101, 400$ [on hold]

Same Mom trying to help daughter here. Once I know how to explain this pattern, can you tell me what I'd call this area of math so I can go re-teach myself? I feel like I could have done this in ...
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1answer
18 views

Growth rate of large sets

Suppose that ${a_k}$ is a real valued increasing sequence such that $$ \sum_{k=1}^{\infty} \frac{1}{a_k} = +\infty ,$$ i.e. $\{a_1,a_2,\ldots\}$ is a large set. If $\lim a_{k+1} - a_{k} = \infty$, ...
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3answers
129 views

The sum of the following infinite series $\frac{4}{20}+\frac{4\cdot 7}{20\cdot 30}+\frac{4\cdot 7\cdot 10}{20\cdot 30 \cdot 40}+\cdots$

The sum of the following infinite series $\displaystyle \frac{4}{20}+\frac{4\cdot 7}{20\cdot 30}+\frac{4\cdot 7\cdot 10}{20\cdot 30 \cdot 40}+\cdots$ $\bf{My\; Try::}$ We can write the given ...
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0answers
24 views

Can I use the Squeeze theorem with sequences of functions?

For example if I know that $f_n(x)\leq g_n(x) \leq h_n(x)$ for all $x$, then can I say that $ \lim_{n \rightarrow \infty} f_n(x)\leq \lim_{n \rightarrow \infty} g_n(x) \leq \lim_{n \rightarrow \infty} ...
5
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1answer
62 views

Strange behavior of infinite products $\prod^{\infty}_{n=1} \ln (1+ \frac{1}{n} )^n$ and $\prod^{\infty}_{n=1} \ln (1+ \frac{1}{n} )^{n+1}$

There are two expressions marking the lower and upper bounds for number $e$: $$\left(1+\frac{1}{n} \right)^n \leq e \leq \left(1+\frac{1}{n} \right)^{n+1}$$ Naturally, I wanted to know if infinite ...
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1answer
27 views

Number of Unique Ranks of High Card in Three Card Brag

Well the game is called Teen Patti in India. Almost similar to Three Card Brag a British game. There are total $16440$ Unique High Card hands are present. (Considering the suit.) Hand $1 = 5$ Heart, ...
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0answers
12 views

About Convergence of a series [on hold]

Is this series convergent? $$\sum_{n=0}^{N-1}\frac{c_{n}^{N}}{c_{N}^{N}n^2}$$ where $c_{n}^{N}$ is coefficient $x^{n}$ in chebyshev polynomial $T_{N}(x)$, i.e. ...
0
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4answers
22 views

Evaluating simple summation

can someone help with this summation. Seems simple, but... I have tried several options but cannot see the rule. $\displaystyle 1-a+a^2-a^3+...a^{2008}-a^{2009}+\frac{a^{2010}}{1+a} {\text{ when}}\ ...
0
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0answers
71 views

Sum of all rational numbers up to infinity [on hold]

Based on the well known sum of natural numbers where $$\sum_{n=1}^\infty n=-\frac{1}{12}$$ Does it make sense to say that the sum of all rational number is zero? How to come with this? Simply take ...
2
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0answers
29 views

Convergence and value of infinite product $\prod^{\infty}_{n=1} n \sin \left( \frac{1}{n} \right)$?

Since the limit $\frac{\sin(x)}{x}=1$ for $x \rightarrow 0$, I wondered about the infinite product: $$\prod^{\infty}_{n=1} n \sin \left( \frac{1}{n} \right)=\sin(1) \cdot 2 \sin\left( \frac{1}{2} ...
0
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1answer
36 views

How to establish convergence and find limit of the sequence$(n+1)^{1/\ln(n+1)}$ [on hold]

How to establish convergence and find limit of the sequence $(n+1)^{1/\ln(n+1)}$. I know its a stupid question but its kinda urgent so please help me out! Edit 1: It is urgent because I have to ...
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0answers
25 views

Absolute Convergence of Infinite Weierstrass Product

I am really stuck on something. I need to show the following: Let $U$ be a domain in $\mathbb{C}$. If $f_n: U \to \mathbb{D}$ are analytic functions satisfying that $\sum |f_n - 1|$ converges ...
2
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2answers
70 views

Taylor expanding $\frac{e^x}{x}$?

How can you taylor expand $$\frac{e^x}{x}$$ Can it be expanded at $x = 0$? Can it be expanded as $x \to 0$?