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

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8 views

How to find a simplified expression for $\binom{1/2}{n}$?

How to find a simplified expression for this specific binomial coefficient? $$\binom{\frac{1}{2}}{n}$$
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1answer
10 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
15 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
15 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
13 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
51 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 + ...
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2answers
23 views

Why is this the closed-form solution for this series?

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
30 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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0answers
22 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
21 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
43 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
42 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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0answers
48 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
87 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
17 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 $\lim a_{k+1} - a_{k} = \infty$, ...
3
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2answers
80 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} ...
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1answer
57 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
23 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
11 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. ...
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4answers
19 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}}\ ...
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0answers
59 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 ...
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0answers
28 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} ...
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1answer
33 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
21 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
68 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 $x \to 0$?
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0answers
28 views

Can you estimate the difference of primes between numerator and denominator?

Let $p_n$ the nth twin prime, it is $p_n$ is a prime number and $2+p_n$ is also a prime. It is well know that Brun's theorem states (unconditionally) that $$\mathcal{B}=\sum_{n\geq ...
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3answers
24 views

How do I decide which convergence criterium to use and what to do if they don't work?

Take $$ \sum_{n=1}^\infty (-1)^nn^{1/n} $$ I just blindly tried (Ratio test) $$ \limsup_{k\to\infty} \left|\frac{a_{k+1}}{a_k}\right| $$ and (Root test) $$ \limsup_{k\to\infty} \sqrt[k]{|a_k|} $$ ...
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1answer
12 views

Check if this is the example of $x$ as a limit point of $C$ but ($x_t$) does not converge to $x$.

Let ($x_t$) be a sequence in a metric space, and let $C$ be the range of ($x_t$). I want to give an example in which $x$ is a limit point of $C$ but ($x_t$) does not converge to $x$. Here's my ...
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1answer
30 views

How to determine if this musical exercise is valid: will the pattern complete?

I'm hoping that math has an answer to a question arising out of a musical exercise. In music terms, the exercise is: Choose two arpeggios (sets of notes) of equal (or roughly equal) span (number ...
13
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1answer
74 views

Pythagorean triplets of the form $a^2+(a+1)^2=c^2$ and the space between them

I was searching for pythagorean triples where $b=a+1$, and I found using a python program I made the first 10 integer solutions: $0^2+1^2=1^2$ $3^2+4^2=5^2$ $20^2+21^2=29^2$ $119^2+120^2=169^2$ ...
0
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1answer
23 views

Limit Evaluation of a Function in the Complex Field

Given the sequence \begin{equation} z_n=\frac{1}{2n\pi}, \quad n \in \mathbb{N} \end{equation} try to evaluate the following limit: \begin{equation} \lim_{z \to z_n} f(z) \end{equation} where ...
1
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0answers
30 views

Derive an inequality using Summation by Parts

Can someone help me to derive the following inequality using Summation by Parts? $a_n$ is a decreasing sequence of positive terms. $$\left|\sum_{k=m+1}^{n+p} a_k \sin kx\right| \le ...
2
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1answer
18 views

Sequential compactness of smooth functions

Suppose I have a sequence $u_n$ of smooth functions on the $N$-dimensional reals. If $\|D^{\alpha}u_n\|_{\infty} \leq C_{\alpha}$ for all multi-indices $\alpha$, then is it possible to deduce that ...
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0answers
20 views

Is the limit of a Dirichlet series to infinity always equal to the constant term?

Given a sequence of complex numbers $(a_n)$ one defines the corresponding Dirichlet series as $$f(s) = \sum_{n = 1}^\infty \frac{a_n}{n^s}.$$ I know that there is some $x_0 \in \mathbb{R} \cup \{\pm ...
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1answer
30 views

A variant of Viète's formula (the 2's replaced by 3's)

I am wondering whether there exists an easy way to evaluate the following infinite product : ...
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0answers
27 views

Find minimum value of $n$ for an integer $A$ such that $A=n^x$,where $n>1$ and $x\geq 1$

How can I calculate sum of a series of function $f(A)$ for $A = 2,3,4,5,6...A$ $f(A)=n$ (such that $n$ is minimum integer such that $A=n^x$ where $n>1$ and $x≥1$ and both n and x are integer) ...
0
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1answer
16 views

What does it mean when it says that the sequence of functions $f_n$ decreases monotonically?

Does it mean that if $x>y$ then $f_n(x)<f_n(y)$ for all $n$ or that for all $x$, $f_{n+1}(x)<f_n(x)$?
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2answers
57 views

How to find the nature of this series?

What tests to use for this series: $$\sum_{m=1}^{\infty}(-1)^{m-1}\left(1+\frac{8}{m}\right)^m$$ I've tried alternating test and ratio test but were inconclusive. Can I apply nth term test to the ...
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3answers
825 views

Summation of a term to infinity

I read through many tutorials but no one mentioned this explicitly. Is the following conversion valid? $$\sum_{k=0}^\infty \frac{k-1}{2^k} = \lim_{n\to \infty} \sum_{k=0}^n \frac{k-1}{2^k}$$ ...
0
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1answer
21 views

Cartesian product to direct sum

I have no idea, how to prove rigorously the corollary from the proposition. I know that i can use the isomorphism $\phi:x_1e_1+...+x_me_m \in \oplus_i^mvect(e_i)\to (x_1e_1,...,x_m e_m) \in ...
3
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2answers
75 views

Can you get a closed-form for $\prod_{p\text{ prime}}\left(\frac{p+1}{p-1}\right)^{\frac{1}{p}}$?

When I use the Taylor expansion series for $$\log(1+x)^{1+x}+\log(1-x)^{1-x}$$ with $x=\frac{1}{p}$, $p$ prime, I believe that I can deduce $$\sum_{p\text{ ...
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1answer
51 views

Given the sequence of functions, $f_1(x):=\sin(x)$ and $f_{n+1}(x):=\sin(f_n(x))$, why $|f_n(x)|\leq f_n(1)$?

I'm analyzing this sequence of functions (for $x\in \Bbb R$): $$\begin{align}f_1(x)&:=\sin(x)\\f_{n+1}(x)&:=\sin(f_n(x))\end{align}$$ to show if it converges uniform or pointwise. My book ...
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2answers
64 views

How do I evaluate a series? [on hold]

In this specific example, I don't understand the steps of evaluating this series: \begin{align} &\frac{12}{n}\left(\left[\sum_{i=1}^n-7\right]+\sum_{i=1}^n\left[\frac{-12}{n}\cdot ...
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0answers
32 views

Is there a specific name for these methods of summation?

When calculating summation of series I use these methods ; Ex: Method One $$U_r=\frac{1}{(r-1)r(r+1)(r+2)}$$ $$U_r=\frac{1}{(r-1)r(r+1)(r+2)}\left[\frac{(r+2)-(r-1)}{3}\right]$$ Then ...
12
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3answers
178 views

How to show the divergence of $\sum\limits_{n=1}^\infty\frac{\sin(\sqrt{n})}{\sqrt{n}}$

The 10 standard tests taught in class are: 1) $n^{th}$ term test for divergence.(Not applicable: $\lim =0$). 2) Geometric Series(Not applicable). 3) Telescoping Series(Not applicable) 4) Integral ...
7
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3answers
92 views

What is the sum of all the natural numbers between $500$ and $1000$.

What is the sum of all the natural numbers between $500$ and $1000$ (extremes included) that are multiples of $2$ but not of $7$?
2
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1answer
19 views

Splitting a sum to find a closed form of $\sum_{n=2}^\infty\frac{n}{n-1}x^{n-1}$

Find a closed form for $$S = \sum_{n=2}^\infty\frac{n}{n-1}x^{n-1}$$ My solution The radius of convergence is $R=1$ and the series does not converge in $\pm 1$. Rewrite the sum as ...
1
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2answers
32 views

What is the correct notation for every nth term in a sequence?

How do I denote every nth term in a sequence? For example, if sequence $C$ contains: $C = \{ 2, 5, 3, 6, 4, 5, ...\}$ And sequence $Q$ contains every 4th term in C: $Q = \{C_{4}, C_{8}, ...
5
votes
1answer
67 views

Find the limit of this sequence

Suppose $$ f_n(x)=\sum_{k=1}^n \frac{\cos(kx)}{k}, $$ and let $a_n=\min_{x \in [0,\pi/2]} f_n(x)$, find $\lim_{n \to\infty} a_n$. I wrote a program and found that the $\arg\min_{x \in [0,\pi/2]} ...