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

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13 views

Analytic representation of Harmonic numbers

As we know, using $$\frac{{{{\ln }^2}\left( {1 - x} \right)}}{{1 - x}} = \sum\limits_{n = 1}^\infty {\left( {H_n^2 - {\zeta _n}\left( 2 \right)} \right){x^n}} = \sum\limits_{n = 1}^\infty {\left( ...
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1answer
17 views

Convergence Proof Help?

Let ($x_n$) and ($y_n$) be given, and define ($z_n$) to be the "shuffled" sequence ($x_1, y_1, x_2, y_2, x_3, y_3,...x_n, y_n$). Prove that ($z_n$) is convergent if and only if ($x_n$) and ($y_n$) are ...
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1answer
52 views

Can the Fibonacci sequence be written as an explicit rule?

When I learned sums and sequences in algebra II with trig I learned about recursive rules and explicit rules. A recursive rule written with the formula of: $$a_n = r * a_{n-1}$$ Or as: $$a_n = ...
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2answers
49 views

How to solve $\sum_{k=1}^n\frac{k}{n^2+k}$?

Can someone show me what is wrong with the expression I got for evaluating $\sum_{k=1}^n\frac{k}{n^2+k}$? Steps: $\sum_{k=1}^n\frac{k}{n^2+k} = \frac{\sum_{k=1}^nk}{\sum_{k=1}^{n}n^2+k} = ...
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3answers
44 views

Showing that a sequence is unbounded

How do I show this sequence is unbounded. ${b_j=j}$ from j=1 to infinity By using the following definition. ${b_j}$ is called bounded if there exist $M>0$ such that $b_j<M$ for all $j\in$ ...
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1answer
17 views

Proving a this sequence converges using epsilon definition

I am not certain how to prove this sequence converge using the epsilon definition Here is the sequence $a_j=\frac{j^2+3}{2j^2-j+9}$ I turned the sequence into a function and for the limit I got $j ...
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0answers
18 views

How to study the nature of this series? [on hold]

Can somebody please help me to study the nature of this series? https://goo.gl/YBYSbM
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3answers
74 views

Evaluate the summation $\sum_{k=1}^{n}{\frac{1}{2k-1}}$

I need to find this sum $$\sum_{k=1}^{n}{\frac{1}{2k-1}}$$ by manipulating the harmonic series. I have been given that $$\sum_{k=1}^{n}{\frac{1}{k}} = \ln(n) + C$$ where $C$ is a constant. I have ...
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2answers
36 views

Root test for convergence: $\displaystyle{\lim_{n\to\infty} (a+bi)^n}$

$$\lim_{n\to\infty} (a+bi)^n$$ where $i$ is the imaginary unit. I'm having trouble with this question. I get to $a+bi$ but I have no clue how to finish it in order to determine if it converges ...
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1answer
18 views

Find the population by the end of the same year…

The population of a type of insect is known to be 200,000 on 1st January in a particular year. Each month, the population increases by 75,000. Find a.) the total population by the end of the same ...
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0answers
31 views

Square root algorithm. Rudin PMA ch.3 problem 17

Fix $\alpha>1$. Take $x_1>\sqrt{\alpha}$ and define $$x_{n+1}=\dfrac{\alpha+x_n}{1+x_n}=x_n+\dfrac{\alpha-x_n^2}{1+x_n}.$$ It's easy to check that $\{x_{2n}\}_{n=1}^{\infty}$ is increasing and ...
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1answer
45 views

To determine the existence and the value of $\lim_{x \to 0} \frac {2^x-1} x$

I would like to somehow firstly show that $$\lim_{x \to 0} \frac {2^x-1} x$$ exists and determine the value of the limit. My first ideas were by Monotone Convergence. I have been able to prove that ...
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1answer
21 views

The Sum of the first five terms of an arithmetic sequence is 65/2… [on hold]

The sum of the first five terms of an arithmetic sequence is 65/2. Also, five times the seventh terms is the same as six times the second term. Find the first term and the common difference of the ...
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1answer
29 views

Approximaing Gamma function

For $c>1$ and $0<\theta<1$, we wish to approximate (upper bound) following Gamma function: $$\int_c^{c\theta}x^{-3}e^{-x}dx $$
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2answers
46 views

Generalizing limits of sums, products, and quotients of sequences to abstract topological spaces?

Introductory real analysis books usually state some properties about limits of sequences of numbers. Suppose $\lim x_n=x$ and $\lim y_n=y$. Then: $\lim (x_n+y_n) = x+y$ $\lim c x_n = cx$, for $c \in ...
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1answer
31 views

Geometric Progression sums and sums of squares

Sum of the first $4$ terms in GP is $30$ and the sum of their squares is $340$. Find the numbers. How do I solve this?
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0answers
24 views

Chance of wining Uno 7 Times in a row

I am Intrigued to determine the odds of winning such a game with 4 players.. and in 7 times consecutively Would be great to have a clear answer on this Thks Phil
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0answers
34 views

Coefficients of the Fourier-Bessel series of $f(x)=x^2$

I'm trying to calculate the expansion of $f : [0,1]\to\mathbb{R}$ given by $f(x)=x^2$ in a Fourier-Bessel series of zeroth order. In that case let $J_0$ be the $0$-th order Bessel function and ...
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0answers
17 views

Find first positive perfect square in polynomial time

I have a quadratic. for example $$1x^2+6884x+3297$$ Is it possible to find the first perfect square in the series in polynomial time where both x and y are whole positive integers. In the above ...
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2answers
50 views

Limiting value of $\frac{x^n e^x}{n!}$ as $n\to\infty$

For the Taylor Series the remainder is of the form $$R_n = \frac{(x-a)^n}{n!} f^{(n)}(\xi) $$ with $a \leq \xi \leq x$ For the series of $e^x$ about $0$ (that is, the Maclaurin series) the remainder ...
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0answers
13 views

Does this property of time dependent sequences have a name?

For $i\in \mathbb{N}$ let $ \chi_i \colon \mathbb{R}^+ \to \mathbb{R}$ be such that for each $t\in \mathbb{R}^+$ we have $\chi_i(t) \in \mathcal{l}_2$ Suppose further that ...
12
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3answers
546 views

Can we add an uncountable number of positive elements, and can this sum be finite?

Can we add an uncountable number of positive elements, and can this sum be finite? I always have trouble understanding mathematical operations when dealing with an uncountable number of elements. ...
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1answer
49 views

Proving that and how $ \frac{1}{n}\sum\limits_{p\le n}\lfloor n/p \rfloor - \sum\limits_{p\le n} 1/p $ approaches $0$

Let $p$ denote a generic prime number. By Mertens' second theorem, the sequence $$\sum\limits_{\ p \le n} \frac1p - \log\log n$$ converges to the Meissel-Mertens constant $M\approx 0.2614972$. Now let ...
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1answer
39 views

limit of sum $\dfrac{(-1)^{n-1}}{2^{2n-1}}$

What is: $$\sum^{\infty}_{n=1}\dfrac{(-1)^{n-1}}{2^{2n-1}}$$ I have done a Leibniz convergence test and proved that this series converges, but I do not know how to find the limit. Any suggestions?
2
votes
2answers
36 views

What function is this? $\sum_{k=0}^\infty \frac{2^{2k}z^{2k-1}}{(2k)!}$ [on hold]

I am interested to know what function represents the following series: $$\sum_{k=0}^\infty \frac{2^{2k}z^{2k-1}}{(2k)!}$$
0
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2answers
29 views

Method of finite differences solutions help?

For homework we were given this sequence: -2 8 27 85 260 ____ 2365 And asked to find the number in the blank. Well, I got the ...
0
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2answers
33 views

Sequences and Series - Find the value of n for which…

I am having some difficulty trying to solve this question. I have been given this question - Find the correct value of the letter n for which Xn = 5n - 2 and ...
3
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0answers
37 views

Is there anything known about the zeros of $\displaystyle \sum_{n=1}^{\infty} \left(\frac{1}{\rho_n^s} +\frac{1}{\overline{\rho_n}^s}\right)$?

Assuming the RH and $\rho_n =\frac12 + \gamma_n i$ being the n-th non-trivial zero of $\zeta(s)$, then numerical evidence suggests that: $$f(s) :=\displaystyle \sum_{n=1}^{\infty} ...
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3answers
101 views

How to find $\lim_{n\to\infty}\left(\frac{\pi^2}{6}-\sum_{k=1}^n\frac{1}{k^2}\right)n$?

How to find $\lim_{n\to\infty}\left(\frac{\pi^2}{6}-\sum_{k=1}^n\frac{1}{k^2}\right)n$? It is well-known that $\lim_{n\to\infty}\sum_{k=1}^n\frac{1}{k^2}=\frac{\pi^2}{6}$, so ...
5
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2answers
121 views

What is the sum of this series? $\sum_{n=1}^\infty\frac{\zeta(1-n)(-1)^{n+1}}{2^{n-1}}$

I want to know what is the sum of this series: $$\sum_{n=1}^\infty\frac{\zeta(1-n)(-1)^{n+1}}{2^{n-1}}$$ $B_n$ are the Bernoulli numbers. Mathematica does not help.
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1answer
51 views

What is the sum of this series? $\sum_{n=1}^\infty\frac{2^n (-1)^{n+1} B_n}{n}$ [duplicate]

I want to know what is the sum of this series: $$\sum_{n=1}^\infty\frac{2^n (-1)^{n+1} B_n}{n}$$ Mathematica does not help.
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2answers
135 views

Solve this integral:$\int_0^\infty\dfrac{\arctan x}{x(x^2+1)}\mathrm dx$

I occasionally found that $\displaystyle\int_0^{\frac{\pi}{2}}\dfrac{x}{\tan x}=\dfrac{\pi}{2}\ln 2$. I tried that $$\int_0^{\frac{\pi}{2}}\dfrac{x}{\tan x}=\int_0^{\frac{\pi}{2}}x \ \mathrm ...
2
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1answer
56 views

Algorithm for computing square roots.

Fix a positive number $\alpha$. Choose $x_1>\sqrt{\alpha}$ and define $x_2, x_3, x_4, \dots$ by the recursion formula $$x_{n+1}=\frac{1}{2}\left(x_n+\frac{\alpha}{x_n}\right).$$It's easy to check ...
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1answer
37 views

infinite series question involving sigma

how to find out the sum of infinite series question $$\displaystyle\mathop{\sum^{\infty}\sum^{\infty}\sum^{\infty}}_{i=0\ j=0\ k=0\ i\neq j\neq k}\frac{1}{3^i}\cdot \frac{1}{3^j}\cdot \frac{1}{3^k}$$ ...
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0answers
62 views

Limit of recursive sequence $x_{n+1}=\frac1n(x_1+2x_2+3x_3+…+nx_n)$

I was trying to solve the following limit but I just can't get it: Let $x_1 = a$, $a>0$, and, for every $n \in \mathbb{N}$, $$x_{n+1}=\frac{x_1+2x_2+3x_3+...+nx_n}{n}.$$ Determine : ...
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1answer
42 views

Interchanging the order of summation for a particular double series.

I suspect, based on numerical approximation, that $$\sum_{n=1}^{\infty}\sum_{m=1}^{\infty}\frac{(-1)^m}{2m+1}\frac{1}{n^{2m+1}} = ...
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4answers
73 views

Convergence or divergence of the series $\sum\limits_{n = 1}^{\infty} \sin(\pi/n)$

Let $ u_{n} = \sin \! \left( \dfrac{\pi}{n} \right) $, where $ n \in \Bbb{N} $, and consider the series $ \displaystyle \sum_{n = 1}^{\infty} u_{n} $. Which of the following is/are true? (a) $ ...
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2answers
43 views

Can absolute value functions be moved like this?

If I have an expression that looks like $|x-a_1| + |x-a_2| + |x-a_3| + ... + |x-a_n|$ Is it the same as doing $|nx - \sum_{i=1}^{n}a_i|$
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1answer
26 views

Identical Summation and Integration of specific functions

A strange coincidence which I discovered recently, is that $$\int_{-\infty}^{\infty}{\tan^{-1}{\frac{1}{(x-\alpha)^2+\frac{3}{4}}}}dx = ...
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1answer
59 views

What is the limit of the sum of “last half” part of harmonic series?

I'm looking for the limit of this sum: $\frac{1}{\left\lceil\frac{n}{2}\right\rceil+1}+\frac{1}{\left\lceil\frac{n}{2}\right\rceil+2}+\frac{1}{\left\lceil\frac{n}{2}\right\rceil+3}+\cdots+\frac{1}{n}$ ...
11
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3answers
152 views

Show that $(1+\frac{1}{n})^n+\frac{1}{n}$ is eventually increasing

I would like to find a way to show that the sequence $a_n=\big(1+\frac{1}{n}\big)^n+\frac{1}{n}$ is eventually increasing. $\hspace{.3 in}$(Numerical evidence suggests that $a_n<a_{n+1}$ for ...
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2answers
52 views

Limit of a partial sum [on hold]

I want to find the limit $$\lim_{n\to \infty}\sum_{i=1}^n \frac{1}{n+i}$$ I tried this. But I am not able to do it. Can anyone please help how to proceed?
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+50

Name for kind of big O notation with leading coefficient

Context: As known the big O notation $O(f(n))$ describes a function $g(n)$ such that there is a constant $C \ge 0$ with $\limsup_{n\to\infty} \left|\frac{g(n)}{f(n)}\right| \le C$ (I assume that ...
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1answer
37 views

How many ways can an integer $i$ appearing in a sequence with multiplicty at least $j$, be minimal

Let us construct an integer sequence of length $n$, where the integers are chosen from $\{1, 2, ..., k\}$, with i.i.d. uniform probability $\frac{1}{k}$. I want to compute the probability ($p_{ij}$) ...
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0answers
18 views

Comparison of bivariate generating functions

Suppose we have two bivariate ordinary generating functions describing two integer sequences which have indicies $a,b$ and $c, d$ respectively. Is there a straightforward way to determine, from ...
2
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2answers
26 views

A analytic representation of q- rational series

Using Mathematica, we can find $$\sum\limits_{n = 1}^\infty {\frac{{{{\left( {1 - q} \right)}^2}{q^n}}}{{\left( {1 - {q^n}} \right)\left( {1 - {q^{n + 1}}} \right)}}} = q,\;q \in \left( {0,1} ...
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0answers
42 views

Problem 14 from Baby Rudin chapter 3

Let $\{s_n\}$ is a complex sequence, define its arithmetic means $\sigma_n$ by $$\sigma_n=\dfrac{s_0+s_1+\dots+s_n}{n+1},\quad a_n=s_n-s_{n-1} \quad\text{for} \quad n\geqslant 1$$ Assume ...
26
votes
6answers
2k views

A way to calculate e?

Define three sequences: The first sequence is $$n^n: 1,\ 4,\ 27,\ 256,\ 3125,\ 46656, \ldots$$ The second sequence is that of the ratios between adjacent members of the first series, or ...
1
vote
1answer
145 views

Why is $1+2+3+\cdots = 0 $? [duplicate]

I had seen this result a while back in a Numberphile video: $1+2+3+\cdots = -\frac{1}{12}$ I was trying to prove the same result using a different method when I accidently proved that the sum was ...
2
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0answers
35 views

convergence of general harmonic series

My question is about determining the convergence of a general harmonic series using the integral test. According to the following resource: pg.32 , we can see that for a general harmonic series ...