For questions about recurrence relations, convergence tests, and identifying sequences.

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1answer
22 views

What to know before solving sequence and series problems?

Talking about How to solve the sequence: $87, 89, 95, 107, ?, 157$ for example, I read the hint: The difference between each term goes like this: 2,6,12. Can you notice any pattern? Based on it, ...
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4answers
45 views

Find the Limit of the sequence $\frac{(2^n)(n!)}{(2n+1)!}$

Find the Limit of the sequence $\frac{(2^n)(n!)}{(2n+1)!}$ I'm not sure how to approach this problem. I tried the squeeze method, but could not figure it out.
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1answer
36 views

How to resolve the issue of two sequences converging to zero for $n, m \to \infty$?

My question is motivated by the following exercise in probability theory: Let $X_n \to X$ in probability and $X_n \geq Y$ a.s. Show that $X \geq Y$ a.s. I noticed that for all $n, m \in ...
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0answers
16 views

Tighter upper bounds with ratios of powers of norms

This question arises in concentration or sparsity measures for finite sequences. Given $x\in \mathbb{R}^K$ and $1 \le r < s$, i try to find a tight upper bound for $$\psi_{r,s}(x) = \frac{\sum_1^K ...
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1answer
30 views

Finding the series for $(\ln(1+z))^2$

So, I'm supposed to use product of infinite series methods to find the series for $(\ln(1+z))^2$. I'm given that the answer has the form $$z^2 \sum_{l=0}^{\infty} c_l z^l$$ and I'm given that the ...
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1answer
40 views

If $\displaystyle\lim_{n\to\infty}|x_n|^{1/n} = L < 1$ then $\displaystyle\lim_{n\to\infty}|x_n|=0$. [on hold]

Let $(x_n)$ be a sequence of real numbers such that $\displaystyle\lim_{n\to\infty}|x_n|^{1/n} = L < 1$. Show that $\displaystyle\lim_{n\to\infty}|x_n|=0$. Remark: Not use that exist $0<r<1$ ...
3
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1answer
32 views

Convergence of fixed-point iteration for $p$ times continuously differentiable function

I am stuck at this problem: Let $\alpha\in\Bbb{R}$ be some number that satisfies $g(\alpha)=\alpha$ for some function $g$ that is $p$ times continuously differentiable on some neighborhood of ...
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2answers
41 views

Terms of a certain recurrence

Let $a_1, a_2\dots $ be a sequence of reals such that $a_1 = a_2 = 1$, and $$a_{n + 2} = \frac{a_{n + 1}^3 + 1}{a_n}$$ for $n \ge 1$. It appears to be the case that all of these values are integers. ...
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0answers
44 views

What are some tricks to solve Progressions quickly?

Recently while solving few questions related to Progressions(specifically, A.P.), I realized one thing that in question like, "Find the sum of following series" and suppose the terms are up to ...
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0answers
45 views

Can someone answer and explain the missing term of this sequence problem? [on hold]

$8,28,?,14,14,6,17,3,...$ Which one is the answer and why? a) $9$ b) $11$ c) $10$ d) $12$
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3answers
64 views

Showing that the sequence $x_n=\frac{1}{3}x_{n-1}(4+x_{n-1}^3)$ where $x_0=-0.5$ quadratically converges

I am stuck at a point in solving this problem: Show that the sequence defined by: For all $n\in\mathbb{N}, x_n = \begin{cases} -\frac{1}{2}, & \text{if $n=0$} \\ ...
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3answers
61 views

find the complex number $z^4$

Let $z = a + bi$ be the complex number with $|z| = 5$ and $b > 0$ such that the distance between $(1 + 2i)z^3$ and $z^5$ is maximized, and let $z^4 = c + di$. Find $c+d$. I got that the ...
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2answers
68 views

pointwise almost everywhere convergent subsequence of $\{\sin (nx)\}$ [on hold]

Can you prove or disprove that the sequence $\{\sin (nx)\}$ has a pointwise convergent almost everywhere subsequence with respect to the Lebesgue measure on $\mathbb{R}$ ?
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1answer
43 views

What is the next value in this sequence? [on hold]

16 104 572 2574 9009 22522.5 ?
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0answers
26 views

FRACTRAN for natural numbers

Is there a simple analogue of FRACTRAN that maps a natural number to a natural number, instead of mapping a list of fractions to a natural number? One could use Gödel encoding to translate FRACTRAN ...
1
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1answer
36 views

What is the definition of the absolute convergence of an infinite product $(1+a_1)(1+a_2)(1+a_3)\cdots$?

For an infinite product $(1+a_1)(1+a_2)(1+a_3)\cdots$, whats the definition of convergence and absolute convergence? Why the absolute convergence corresponding to the absolute convergence of sum ...
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2answers
29 views

What is the difference of the greatest of the limits $\overline{\lim}_{n\to \infty}$ and the least of the limits $\underline{\lim}_{n\to \infty}$?? [on hold]

What are exactly these three limits for an infinite series $x_n$? $$\overline{\lim_{n\to \infty}} x_n$$ $$\underline{\lim}_{n\to \infty} x_n$$ $$\lim_{n\to \infty} x_n$$ Can they be different from ...
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1answer
37 views

Calculat sums of the form $\lim_{n\rightarrow\infty}\sum_{i=0}^{n}\left(\dfrac{i}{n}\right)^{f(n)}$

Problem: calculate the sums of the form: $$\lim_{n\rightarrow\infty}\sum_{i=0}^{n}\left(\dfrac{i}{n}\right)^{f(n)}$$ Inspiration: one problem lets us prove that ...
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2answers
34 views

Trying to understand why 2 times the sum of consecutive integers from 0 to n is equal to n times n+1

I am sorry if this question ends up being a duplicate, as I am having a bit of a challenge explaining it to myself well enough to know how to query it. There is a Facebook meme that has been ...
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0answers
19 views

Intriguing Poisson sum with hyperbolic function

I've been playing with lots of Poisson sums lately, and I thought this one to be interesting:\ $$\sum_{k\in\mathbb{Z}}\left(\frac{1}{(k+x)\sinh{(k+x)\pi q}}-\frac{1}{\pi q (k+x)^2}\right)$$I want to ...
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6answers
73 views

For which values of $x$ does this series converge?

For which values of $x$ does the series presented below converge? $$\sum_{n=1}^{+\infty}\frac{x^n(1-x^n)}{n}$$ Neither the root test nor the ratio test is of much help - I've tried for ...
2
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1answer
40 views

Finding a general term for the sequence

Find a general term in simplest form for the sequence: $$2, 1, -4, 7, -10, 13, -16$$ This is what I tried: $a_n = a_1 + (n - 1)\cdot d$ $a_n = 2 + (n-1)\cdot 3$ $a_n = (-1)^n (2+(n-1)\cdot 3)$ which ...
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0answers
12 views

Double Sum of Series Unchanged When Each Term Scaled

Can the following ever hold: $$\sum_{i}\sum_{j}(-1)^ja_{i,j}=\sum_{i}i(b_{i})\sum_{j}(-1)^ja_{i,j}$$ where $b_{i}>1/i$ for all $i$? What if you tack on the fact that ...
3
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1answer
39 views

Sum involving the number of zeros of $k$

I'm currently interested in sums involving digit-functions. Especially, I'd like to calculate the following sum: $$ s=\sum_{k=1}^{\infty}{\frac{a_0(k)}{k\left(k+1\right)}} $$ Where $a_0(k)$ is the ...
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2answers
25 views

Series And Sequences Question

Can someone help show me what I did wrong? The question is "Find the sum of the first ten terms in this geometric series: $-5, 10, -20, \ldots$ I plugged it into this equation: $S_n = a(r^n ...
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2answers
63 views

Help needed with the integral of an infinite series

Can you please help me with the integral of this series? I came across it in a signal processing paper and haven't been able to figure out the solution myself. $$ ...
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0answers
33 views

Proving an inequality involving discrete variables

I'm trying to show that the following inequality holds $$ \frac{1-x^{n}}{1-x^{n+1}}\geq\frac{\sum_{i=0}^{n-2}x^{i}(1-x_{1}^{n-(i+1)})}{\sum_{i=0}^{n-1}x^{i}(1-x_{1}^{n-i})}, $$ where $n$ is a ...
2
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1answer
54 views

Notion of fixed point for the sequence $x_{n+1}=f_n(x_n)$

Let $(X,d)$ be a complete metric space and let $f_n:X\to X,\ n\in \mathbb{N}$ be a sequence of contractions. Is there any notion of fixed point for the following sequence? $$x_{n+1}=f_n(x_n),\ ...
1
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1answer
50 views

Rearranging a series' terms

So i am asked to rearranje the terms in this series: $$ \sum_{i=1}^\infty \frac{(-1)^{n+1}}{n} = 1- \frac 12 +\frac13-\frac14+... $$ so that the sum of the series is equal to 0. I've seen the ...
2
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1answer
47 views

Find a constant $C$ such that $ \Bigg| \frac{\prod_{i=0}^{k-1} (n-i) }{n^k} - 1 \Bigg| \leq \frac{C}{n}, \forall k \leq n$

Consider the following: $$ \Bigg| \dfrac{\prod_{i=0}^{k-1} (n-i) }{n^k} - 1 \Bigg| \leq \frac{C}{n}, \forall k \leq n $$ How to find an expression for $C$ independent of $k$ and thus $n$? It arises ...
2
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2answers
50 views

How to show that the following series converges to 1

Let $f$ be a function on $\mathbb{R}$, non-zero only on $[0,2)$. In particular $f(x)=1,x\in[0,1]$ and decreasing to zero, starting from $x=1$. Let $g(x)=f(x)-f(2x)$. Show that $$\sum_{j=0}^\infty ...
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2answers
68 views

The Laurent Series of $\dfrac{e^z}{z^2-1}$

The Laurent Series of $\dfrac{e^z}{z^2-1}$ At $z=1$ As we seek for powers of $z-1$, note that: $$e^z=e\cdot e^{z-1}=e(1+(z-1)+\dfrac{(z-1)^2}{2!}+\dfrac{(z-1)^3}{3!}+...)$$ So: ...
3
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1answer
43 views

Does this sequence of functions converge uniformly?

So the questions says, let $a_n$ be a sequences of real numbers such that $\limsup |a_n| = 0$. Let $X = [0, 1]$ and for each $n \in \mathbb{N}$ the function $\space$ $f_n :$ $X \mapsto ...
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0answers
14 views

What are the coefficients of Modified Fourier Bessel series?

What are the co-efficients of Modified Fourier Bessel series? $$\phi g(r,z,Φ) = \sum_{v=1}^∞ \sum_{m=1}^∞ \sin\left(\frac{mπz}{L}\right) I_v \left(\frac{mπr}{L}\right)\left(A_{vm} \cos(vΦ) + B_{vm} ...
3
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1answer
55 views

Proving that a sequence converges or diverges [on hold]

Prove or disprove that there is a sequence $n_k$ of positive integers (that is not constant) such that both $\cos(2n_k!)$ and $\sin(2n_k!)$ converge. I think that the series diverges but I am not ...
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1answer
136 views

An Infinite series I

By decompising fractions one can show that \begin{align} \sum_{n=1}^{\infty} \frac{1}{n \, (n+1)^{2} \, (n+3)} = \frac{65}{72} - \frac{\zeta(2)}{2}. \end{align} The fraction can also be seen in the ...
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1answer
37 views

$ |x_{n} - y_{n}| < \frac{1}{n} \Rightarrow |x'_{n} - y'_{n}| < \frac{1}{n}$

This came from a proof on uniform continuity theorem. My textbook claims that if sequence $(x_j)$ and $(y_j)$ on a compact set D have the condition that $ |x_n - y_n| < \frac{1}{n}$, and $(x'_j)$, ...
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3answers
106 views

Find the sum of series $\sum_{n=2}^\infty\frac{1}{n(n+1)^2(n+2)}$

How to find the sum of series $\sum_{n=2}^\infty\frac{1}{n(n+1)^2(n+2)}$ in the formal way? Numerically its value is $\approx 0.0217326$ and the partial sum formula contains the first derivative of ...
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1answer
18 views

Finding the bounds for a truncation error

I have two series, $S$ and $T$ which approximate $\pi$ such that $$S_n = 4 \sum_{i=1}^n \cfrac{-1^{i+1}}{2i-1}$$ and $$T_n = \Big(12 \sum_{i=1}^n \cfrac{-1^{1+i}}{k^2} \Big) ^{\frac{1}{2}}$$ It is ...
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0answers
12 views

What does “empirical error” mean in this context?

I recently sat an exam for computational mathematics. The question asked for us to: "Write the empirical error in $\mathcal{O}(n^{-p})$ where $p$ is some integer" We were given a series $$S = 4(1 - ...
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1answer
37 views

Is this correct for Rudin exercise 3.7? Prove the series is convergent

This is Baby Rudin exercise 7 of Chapter 3. Prove that the convergence of $\sum{a_n}$ implies the convergence of $\sum{\sqrt {a_k} \over k}$ if $a_n \le 0$. Proof: I will attempt to show that the ...
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5answers
136 views

Show that $(1+p/n)^n$ is a Cauchy sequence for arbitrary $p$

It is a generalization of this question. I am looking for a similar derivation as in here. Can we prove that $(1+p/n)^n$ is a Cauchy sequence for any $p \in [a, b]$ by showing that $$ \Bigg| \left( ...
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2answers
34 views

How to show that this interesting difference of products is $O \left( \frac{1}{n^2} \right) $

Let $k \leq n$. Consider the following difference of products: $$ \prod_{i=1}^{k-1} \left( 1 - \frac{i}{n+1} \right) - \prod_{i=1}^{k-1} \left( 1 - \frac{i}{n} \right)$$ For $n=1,2,3$, this is ...
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2answers
58 views

What textbooks should I use for Trigonometry and Calculus? My basics are terrible.

I need help really bad. I have a paper coming up in two months and all topics require at least basic if not intermediate understanding in trigonometry and calculus. I don't know how I got so far - by ...
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1answer
25 views

Summation of infinite series, where difference in consecutive denominator forms an A.P.

What is the sum of an infinite series where each term can be written as $\frac{p}{q}$, where p=1 always the difference between 2 consecutive denominators forms an A.P. For example $\dfrac{1}{2}$, ...
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2answers
12 views

On the existence of a particular type of real sequence of functions

Does there exist a sequence of real valued functions $\{f_n\}$ with domain $\mathbb R$ which is uniformly convergent ( on some subset of $\mathbb R$ ) to a continuous function and such that each $f_n$ ...
2
votes
2answers
47 views

What's special about the cauchy product?

I am reading chapter 3 in Rudin's Principles of Mathematical Analysis. In it, Rudin defines the Cauchy product of two series. That is, given $\sum a_n$ and $\sum b_n$, $c_n=\sum_{k=0}^n a_k b_{n-k}$ ...
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2answers
37 views

Sum of Converging sequence

I'm given this sequence where it goes: $$ 1,\; \frac1a,\; \frac1{a(a+b)},\; \frac1{a^2(a+b)},\; \frac1{a^2(a+b)^2}, \frac1{a^3(a+b)^2}, \dotsc $$ where $a$ and $b$ are any positive integers How ...
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1answer
44 views

Stadium Seating - Geometric Sequences

A circular stadium consists of sections as illustrated, with aisles in between. The diagram show the tiers of concrete steps for the final section, Section K. Seats are to be place along every step, ...
1
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1answer
33 views

Asymptotic Estimates for a Strange Sequence

Let $a_0=1$. For each positive integer $i$, let $a_i=a_{i-1}+b_i$, where $b_i$ is the smallest element of the set $\{a_0,a_1,\ldots,a_{i-1}\}$ that is at least $i$. The sequence ...