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

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2
votes
3answers
83 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 ...
0
votes
2answers
45 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 ...
-1
votes
1answer
19 views

Find the population by the end of the same year… [on hold]

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 ...
-1
votes
0answers
33 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 ...
0
votes
2answers
56 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 ...
-1
votes
1answer
23 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 ...
1
vote
1answer
41 views

Approximaing Gamma function

For $c>1$ and $0<\theta<1$, we wish to approximate (upper bound) following Gamma function: $$\int_c^{c\theta}x^{-3}e^{-x}dx $$
1
vote
2answers
68 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. 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 ...
0
votes
1answer
34 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?
0
votes
0answers
25 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
2
votes
1answer
89 views

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 ...
1
vote
0answers
22 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 ...
4
votes
2answers
53 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 ...
0
votes
0answers
15 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
votes
4answers
587 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. ...
5
votes
1answer
54 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 ...
1
vote
1answer
40 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
votes
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
votes
2answers
34 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
votes
0answers
39 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} ...
3
votes
3answers
110 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
votes
2answers
124 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.
0
votes
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.
1
vote
2answers
150 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
votes
1answer
57 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 ...
-1
votes
1answer
40 views

infinite series question involving sigma

how to find out the sum of infinite series question $$\displaystyle\mathop{\sum^{\infty}\sum^{\infty}\sum^{\infty}}_{i=0\ j=0\ k=0\ i\neq j\neq k}\frac{1}{3^i}\cdot \frac{1}{3^j}\cdot \frac{1}{3^k}$$ ...
1
vote
0answers
66 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 : ...
1
vote
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}} = ...
0
votes
4answers
77 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) $ ...
0
votes
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 $|nx - \sum_{i=1}^{n}a_i|$
1
vote
1answer
27 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 = ...
1
vote
1answer
62 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
votes
3answers
159 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 ...
1
vote
2answers
53 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?
2
votes
0answers
43 views
+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 ...
1
vote
1answer
42 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 ($p_{ij}$) ...
1
vote
0answers
20 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
votes
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} ...
1
vote
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 ...
28
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
151 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
votes
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 ...
2
votes
6answers
134 views

Find the limit of the sequence $(4^n)/((2n)!)$

How to find $$\displaystyle \lim_{n\to\infty} \frac{4^n}{(2n)!}$$ Can I use l'Hopital's rule?
1
vote
1answer
25 views

Dirichlet theorem

Can anyone give a simple number theory proof for the Dirichlet theorem? Statement of Dirichlet theorem:given any two numbers a and b whose g.c.d is 1,Prove that infinitely many primes exist in the ...
2
votes
2answers
62 views

How can I solve $\sum_{n=1}^{\infty} \frac{8}{(4n-1)(4n-3)}$ step by step? [on hold]

$$\sum_{n=1}^{\infty} \frac{8}{(4n-1)(4n-3)}$$ First off I know the answer to this question, but I do not know how to actually get the answer I do know some basic sums like that one. Can someone ...
0
votes
1answer
68 views

Prove sequence ${x_{n + 1}} = \sin {x_n},{x_1} = 1 $ has a limit

Prove that the sequence defined by $${x_{n + 1}} = \sin {x_n},\ {x_1} = 1$$ has a limit. Ok, I want to prove by Weierstrass: This sequence is monotonically decreasing Sequence is bounded ...
0
votes
1answer
59 views

high order (infinite series)

This question, I have made but there was no answer, so I will try again. If we have the sums $f(n) = 1^{59} + 2^{59} + 3^{59} + \cdots + (10^n)^{59}$ and $g(n) = 1^{5} + 2^{5} + 3^{5} + \cdots + ...
3
votes
2answers
48 views

Calculating in closed form another digamma alternating series

Is there any clever way of finish it fastly? $$\sum _{n=1}^{\infty } (-1)^{n+1} \left(\psi ^{(0)}\left(\frac{5}{8}+\frac{3 n}{8}\right)-\psi ^{(0)}\left(\frac{1}{8}+\frac{3 n}{8}\right)\right)$$ ...
1
vote
2answers
55 views

Explicit formula for recursive sequence.

Consider the series defined by $$P_n=(P_{n-1}-a)b\ .$$ $$P_0=c$$ Basically the number sequence $P_n$ represents the current Principal balance of a debt and constants $a$ and $b$ are constants or ...