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

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2answers
61 views

Rearrangment of convergent series

Consider the convergent series $$1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\frac{1}{5}-\frac{1}{6}+\cdots \tag{$*$}$$ and one of its rearrangments ...
29
votes
8answers
2k views

Can a sequence have infinitely many limits among its subsequences?

Suppose we have an extended real (countably) infinite sequence $(x_n)$. Then consider all of its possible subsequences $(x_{n_k})$. We could then consider the set $$A = \{a\in ...
2
votes
3answers
54 views

Deducing $\sum_{r=1}^{n}r$ from sine summation formula

We know the famous formula $$\sum_{r=1}^{n}\sin r\theta=\sin \frac{n\theta}{2}\csc\frac{\theta}{2}\sin\frac{(n+1)\theta}{2}\ .$$ I have come across a question that use the above result to find ...
2
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0answers
48 views

Optimizing over an infinite set of variables

This may be a very basic question, but it's been a while since I did any optimization. Suppose I have a sequence $(x_i)$, $i=1,2,\ldots$ in the $\ell^2$ space and the following optimization problem: ...
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0answers
53 views

Question re Hold'em Hands with Identical Ranks [duplicate]

In a prior posting regarding a poker session consisting of 142 hands I did not clearly state my question resulting in a misinterpretation. Let me try again. Out of the 142 hands dealt, there were 38 ...
0
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1answer
369 views

Recursive Function for Pyramid-Scheme

consider a group which has 1 user. each month, every user can bring another user to join the group. the user that has been joined for 3 months, should leave the group. calculate the total membership ...
3
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1answer
78 views

Can we say $(\prod_{n=1}^{\infty}n)^2=(\prod_{n=1}^{\infty}n^2)$

Can we say $$(\prod_{n=1}^{\infty}n)^2=(\prod_{n=1}^{\infty}n^2)$$ It works fine if things are finite, does it hold in $n$ goes to infinity?
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6answers
3k views

Is this solution mathematically “legal”?

I have the sequence $$ a_n = \frac{n \cos n}{n^2 + 1} $$ and I'm trying to evaluate the limit of $a_n$ as $n\to\infty$ $$ \begin{align*} \lim_{n\to\infty}a_n&= \lim_{n\to\infty}\frac{n \cos n}{n^2 ...
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4answers
269 views

How to show this limit is zero?

Is it true that the following limit is zero as $n$ goes to infinity for all positive integers $k$? If so, how to prove it? $$n\left[(n-1)^{-\frac{1}{k}}-n^{-\frac{1}{k}}\right]$$
1
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3answers
115 views

Sequence defined by $s_{n+1}=\sqrt{s_{n}s_{n+2}}$

Let $(s_n)$ be the sequence defined by: $$ s_0,s_1\in \mathbb{R}^{+},\quad \forall n\in \mathbb{N};\quad s_{n+1}=\sqrt{s_{n}s_{n+2}} $$ $(s_n)$ is arithmetic sequence $(s_n)$ is ...
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0answers
82 views

Infinite Integration in Limits of Integration

Given the following: $$ u_0 = \int \limits_{ 0 } ^{ 1 } x \, dx , \:\:\: u_1 = \int \limits^{ \int \limits_{ 1/2 } ^{ 1 } x \, dx } _{ \int \limits_{ 0 } ^{ 1/2 } x \, dx } x \,dx , \:\:\: u_2 = \int ...
0
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2answers
30 views

Find the value of x if 1,$\log_{9}(3^{1-x}+2) $, $ \log_{3}(4.3^x-1) $ are in an AP.

We have an AP: 1, $ \log_{9}(3^{1-x}+2) $, $ \log_{3}(4.3^x-1) $ We have to find value of x. $$ d = a_{2} - a_{1} $$ $$ d = log^{(3^{1-x}+2)}_9 - 1 $$ $$ d = log^{(3^{1-x}+2)^{\frac{1}{2}}}_3 - ...
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0answers
38 views

Convergence and limit of a sequence $x_n=1+\frac{x^2_{n-1}}{2},n\ge2,x_1=\frac{3}{8}$

$x_{n+1}-x_n=\frac{x^2_n-2x_n+2}{2}>0$ sequence is increasing. I don't know how to prove that it is bounded. Limit should be $\frac{1}{2}$
1
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1answer
50 views

Sequence identification for these numbers?

Sorry for my English. Can someone help me find the generating formula for these numbers? $$[255,2915,16383,62499,186623,470595,1048575,2125763,3999999,7086243,11943935,19307235,...]$$ All I know is ...
0
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1answer
29 views

A question on convergent series of positive terms

Let $\sum a_n$ be a convergent series of positive terms ; then we know $\lim \inf (na_n)=0$ ; can we derive from here that if $\{a_n\}$ is decreasing , then $\lim (na_n)=0$ ?
6
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1answer
209 views

Infinite Product Representation of $\sin x$

I've recently taken interest in infinite products, and I'm having trouble with a proof I found in this PDF file: "Infinite Products and Elementary Functions": An intermediate step in finding an ...
7
votes
2answers
120 views

Bounding $f'$ in terms of $f$ and $f''$

Assume that $f: \mathbb{R} \to [0,\infty)$ is $C^2$ and $|f''(x)| \leq A$ for all $x$. Show that the inequality $$(f'(x))^2 \le 2Af(x)$$ holds for all $x$. The hint given in the question was, ...
1
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2answers
69 views

Series Approximation How to evaluate $1/3+1/3(1/3)^3+1/5(1/3)^5+…$?

How to evaluate $$\frac13+\frac13(\frac13)^3+\frac15(\frac13)^5+...$$? I faced this particular sum in the website www.toppr.com .And it is given under the heading "Problems on Approximation"...but I ...
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1answer
52 views

Find an expression for the sum of the first n terms of this series

I want to find an expression for the sum of the first n terms of this series: \begin{equation} 5-\frac{5}{2}+\frac{5}{4}-\frac{5}{8}+...+\frac{(-1)^{n-1}5}{2^{n-1}} \end{equation} I have proved that ...
6
votes
3answers
7k views

Prove: Convergent sequences are bounded

I don't understand this one part in the proof for convergent sequences are bounded. Proof: Let $s_n$ be a convergent sequence, and let $\lim s_n = s$. Then taking $\epsilon = 1$ we have: ...
0
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1answer
129 views

Fractional-Recursive Sequence

Here from the fraction set we have a really hard question to be answered...suppose that a sequence is defined as $a_{n} = a_{n-1} - \dfrac 1{a_{n-1}}$, where $a_0$ is given. ...you already know what ...
1
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2answers
46 views

Solve the sequences inequality

If $a_1=1$ and $a_n=a_{n-1}+\dfrac{1}{a_{n-1}}$ for $n≥2$ , then prove that $12 < a_{75} < 15$ ? I have tried solving this by: $$a_{75} - a_1 = ...
1
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2answers
25 views

Convergent series? The pairwise product of the harmonic series with a nonnegative, diminishing to zero, divergent series

Suppose we have a sequence $\{\alpha_n\}$ such that $\alpha_n \ge 0$, $\sum_n \alpha_n = \infty$ and $\alpha_n \rightarrow 0$. Is it true that $\sum_n n^{-1} \, \alpha_n$ converges?
10
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4answers
366 views

explicit formula for recurrence relation $a_{n+1}=2a_n+\frac{1}{a_n}$

For $n\in\mathbb N$, $$a_{n+1}=2a_n+\frac{1}{a_n},\quad a_1=1. $$ Can any one give an explicit formula for all $a_n$? If such an explicit general formula doesn't exist, please explain it. I've tried ...
5
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5answers
132 views

Convergent/divergent series

Is the following series divergent/convergent? ...
0
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1answer
64 views

Partially Identical Hands in Hold'em

I was playing Texas Hold'em at a local cardroom last night keeping a meticulous record of the hands I was dealt. Perhaps I am totally wrong but I thought the occurrences of certain events in this ...
2
votes
4answers
50 views

For what values of $p$ does this series converge?

This is a question we asked on a second semester calculus test. For what values of $p$ does this series converge? $$\sum_{n=1}^{\infty}\frac{\sin(1/n)}{n^p}$$ I believe that it actually can be shown ...
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votes
3answers
73 views

Does the sum of reciprocals of primes congruent to $1 \mod{4}$ diverge?

Let $P$ be the set of primes $p$ greater than $3$ such that $p\equiv1 \pmod{4}$. Does the following sum converge or diverge? $$ \sum_{p\in P}\frac{1}{p} $$
7
votes
2answers
234 views

What is the probability that this harmonic series with randomly chosen signs will converge?

Suppose we fix $p$ between $0$ and $1$ (without loss of generalization, we can assume $p \leq 1/2$). Then suppose we form the series $\sum_n a_n / n$ where the $a_n$ are independent random variables ...
0
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0answers
51 views

How to prove that this series converges. [duplicate]

I want to prove that this series converges: \begin{equation} 1+\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+... \end{equation} I normally use this one as a standard series to test other series for their ...
13
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8answers
1k views

Need to prove the sequence $a_n=1+\frac{1}{2^2}+\frac{1}{3^2}+\cdots+\frac{1}{n^2}$ converges

I need to prove that the sequence $a_n=1+\frac{1}{2^2}+\frac{1}{3^2}+\cdots+\frac{1}{n^2}$ converges. I do not have to find the limit. I have tried to prove it by proving that the sequence is monotone ...
1
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2answers
903 views

Which day of the week will be $100$ days from now

If today is Wednessday, what day of the week will it be $100$ days from now? Going forward $1,8,15,\dots,92,99$ days all result in Wednesday. So the $100$th day will be Thursday. But the solution ...
3
votes
2answers
212 views

Can you use the sum formula for a geometric series starting at any point?

Wherever I see the sum of a infinite geometric series with $|r|<1$ being derived the series always starts at $n = 0$, or $n = 1$, the basic form is $$a + ar + ar^2 + ar^3 + ... $$ And the sum is ...
15
votes
3answers
897 views

How to prove the Fibonacci sum $\sum \limits_{n=0}^{\infty}\frac{F_n}{p^n} = \frac{p}{p^2-p-1}$

We are familiar with the nifty fact that given the Fibonacci series $F_n = 0, 1, 1, 2, 3, 5, 8,\dots$ then $0.0112358\dots\approx 1/89$. In fact, $$\sum_{n=0}^{\infty}\frac{F_n}{10^n} = ...
4
votes
1answer
74 views

An infinite sum based on the mod-parity of Euler's totient function

Let $\bmod( m,k )$ be the remainder when $m$ is divided by $k$: $0,1,\ldots,m{-}1$. Let $\phi(n)$ be Euler's totient function: the number of relatively prime numbers smaller than $n$. So for ...
4
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6answers
316 views

Find a finite sum of perfect squares with alternating signs

How to find the sum of this formula? $$\sum\limits_{k=1}^{2n} (-1)^{k} \cdot k^{2}$$
0
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1answer
31 views

Example of each $\{f_n\}$ Riemann integrable such that $\sum f_n$ converges point-wise to $f$ which is not Riemann-integrable

I am looking for an example of a sequence $\{f_n\}$ of real valued Riemann integrable functions on a closed bounded interval such that $\sum f_n$ converges point-wise to a function $f$ which is not ...
0
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0answers
27 views

Limit Points and Convergence of Sequences.

Let $E \subseteq \mathbb{R}$ (or $\mathbb{C}$). A point $p \in \mathbb{R}$ (or $\mathbb{R}$ ) is called a limit point of $E$, if $\forall \epsilon > 0$, $\exists z \in E$ such that $0 < |z − p| ...
2
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3answers
76 views

Is such a multivariate function the product of two univariate functions?

Let $f: \mathbb{N}^2 \rightarrow \mathbb{R}$ be a function of two variables, $f=f(x,y)$. $f$ has the following property: $$ \sum_{y\in A} f(x,y) = 0 $$ where sum on $y$ runs over a fixed ...
0
votes
2answers
48 views

If $\,S_n/n\,$ converges to a finite limit $c$, why does $\,\left(S_{n+1}-S_{n}\right)/n\,$ converge to zero?

If we have a sequence $S_n$ and know as a fact that $S_n/n$ converges to some finite limit $c$, why is it true that $(S_{n+1}-S_n)/n$ converges to zero? I could see this for $n+1$ in the denominator ...
0
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0answers
18 views

Flipped Point Spread Function

I was reading on wikipedia about the Lucy-Richardson algorithm and its equivalent iterative function: $$ u^{(t+1)}=u^{(t)}\cdot \Big(\frac{d}{u^{(t)}\otimes p}\otimes \hat{p}\Big) $$ where d is the ...
1
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2answers
47 views

inverting the summation

Let $\{a_n\}$ be a sequence of non-negative real numbers. Then, how can one prove rigoroulsy that $$ \sum_{n=1}^\infty \frac{1}{n^2} \sum_{j=1}^n a_j = \sum_{j=1}^\infty a_j \sum_{n=j}^\infty ...
4
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2answers
141 views

Why doesn't this work for Rudin Exercise 3.8

The problem is 3.8 exercise in baby Rudin: If $ \sum{a_n} $ converges and $\{b_n\}$ is bounded and monotonic, prove that $\sum{a_nb_n}$ converges. Why can't I just do this?: Let $M$ be an ...
2
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2answers
31 views

Analysis Limit- Function Proof

A) For every sequence ($p_n$) in $E$ such that $p_n \not= p$ and $p_n \rightarrow p$ as $n \rightarrow \infty$ we have that $f(p_n) \rightarrow l$ as $n \rightarrow \infty$ . ($E \subseteq ...
0
votes
1answer
34 views

Does the sequence have a uniform limit? How do I show this?

I am having problems with the following exercise, I already did part $(i)$ and $(ii)$, I am having problems with $(iii)$. Exercise: Let $ \large f_n(x)=\left\{ \begin{array}{ll} ...
1
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1answer
44 views

Get a number by doubling and re arranging

Inspired by this meta code golf post The question goes like this: Starting with 1 you can repeatedly perform one of the following two operations: Double the number or Rearrange its digits in any ...
3
votes
1answer
65 views

Proving $ \sum_{k=1}^{\infty}\frac{9}{10^{\frac{k(k+1)}{2}}}=0.90900900090… $ is irrational

Now I am proving the number $$ \sum_{k=1}^{\infty}\frac{9}{10^{\frac{k(k+1)}{2}}}=0.90900900090... $$ is irrational. Here I use a similar method to the proof of e is irrational by Joseph Fourier. ...
1
vote
1answer
39 views

Simple Limit Points and Sequences proof

I'm just starting to learn some real analysis and I was wondering if somebody could help verify and critique my proof. If somebody could also provide a different proof that would also be nice. Thank ...
4
votes
5answers
102 views

Prove that $\sum\frac{n+1}{(n+2)n!}$ converges

Show that $\displaystyle\sum\limits_{n=1}^{\infty}\frac{n+1}{(n+2)n!}$ converges, using the integral test. I noticed that $\displaystyle\sum\frac{n+1}{(n+2)n!} = \sum\frac{(n+1)^2}{(n+2)!}$, but ...
0
votes
5answers
200 views

Help finding the limit of this series $\frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \frac{1}{32} + \cdots$

How can I go about finding the limit of $$\frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \frac{1}{32} + \cdots = \sum_{k = 1}^{\infty} \frac{1}{2^{k+1}}?$$ Could I use the absolute value theorem? I have a ...