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

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2
votes
1answer
44 views

natural sequence

I'm trying to solve this exercise on sequences : $$u_{n+1}=\frac{3u_n+4}{2u_n+3}$$ and $$u_0=1$$ for any natural number 1)Find a and b as $$u_{n+1}=a+\frac{b}{2u_n+3}$$ I've found a=3/2 and b=-1/2. ...
0
votes
1answer
93 views

Calculate The Sum of Series

I have a series like this: $$ \sum\limits_{m=0}^{n-1}\frac{2}{n(2n - 2m -1)} $$ In the end, it should be a function of $n$ only where $m$ should be represented by $n$ somehow, I really cannot ...
2
votes
1answer
62 views

Sum of Residues of $\psi^2(-z)$

Compute the Sum of residues of $f(z) = \psi^2(-z)$, where $\psi(z)$ is the digamma function. There are singularities for $z= 1, 2, 3, \ldots$, i.e. for all natural numbers. But how do I compute the ...
1
vote
4answers
308 views

Sum of a decreasing geometric series of integers

I'm trying to calculate the sum of integers that are being divided by two (applying the floor function if necessary): $n\mapsto \lfloor \frac{n}{2}\rfloor$. Let ...
37
votes
1answer
588 views

Does my “Prime Factor Look-and-Say” sequence always end?

I'm trying to create a challenge for PP&CG where the object will be to find the longest sequence in a given time, but I'm worried that there may be an infinite sequence that will ruin things. The ...
1
vote
1answer
110 views

find $\sum_{i=1}^\infty \frac{1}{n3^n}$

How to find $$\sum_{i=1}^\infty \frac{1}{n3^n}$$ Don't know how to start, any hints A rigorous proof is also welcome
4
votes
4answers
120 views

$\lim_{n \to \infty}\frac{n^{100}}{2^n+n^2+5}$

$$\lim_{n \to \infty}\frac{n^{100}}{2^n+n^2+5}$$ I do not know how to handle $n^{100}$ I thought that $2^n$>$n^{100}$ and therefore the $$0=\lim_{n \to \infty}\frac{n^{100}}{3\cdot2^n}\leq\lim_{n ...
2
votes
1answer
60 views

Without using logarithm rules, how to calculate the limit $\lim_{n\to\infty} R(n)^n$ for a rational function $R$?

Well, without using logarithm rules, how to calculate this limit? $$\lim _{n\to \infty }\left(\frac{n^2+8n\:-1}{n^2-4n-5}\right)^n$$ I can't find any more "nice" presentation of this fraction, and ...
0
votes
0answers
48 views

Prove that series is convergent

Let $\displaystyle \sum_{n=1}^{\infty} a_n$ be convergent series of complex terms. Prove there exist unbounded sequance $(b_n)^{\infty}_{n=1}$ of positive terms such that series $\displaystyle ...
4
votes
2answers
108 views

Calculating Harmonic Sums with residues.

Evaluate: $$\sum_{n=1}^{\infty} \frac{H_n}{(n+1)^2}$$ A user stated: "most of the time sum up the residues of $(\gamma+\psi(z))^2\cdot r(z)$. To determine the residues, just expand the digamma ...
2
votes
0answers
98 views

Evaluating a sum $-\zeta'(2)$

Is it possible to obtain any closed-form expression for the infinite sum $$\sum_{n=1}^{\infty}\frac{\log(n)}{n^{2}}$$ by Residue calculus? My thought was to try to integrate $$f(z) ...
0
votes
1answer
51 views

Prove $ \lim_{n\to+\infty}\int^{-\alpha}_{-1}|f_n(t)+1|dt=0$

Suppose $E$ a vector space of continuous function from $[-1,1]$ to $\mathbb{C}$, we define the norm: $$||f||_1= \displaystyle\int^1_{-1}|f(t)|dt$$ and we define a sequence such as: $$ f_n(t)= ...
0
votes
2answers
30 views

How find $n\in\mathbb{N}$ such that ${S_1} < {S_2}$?

Let ${S_1} = \sum\limits_{k = 1}^{4{n^2}} {\frac{1}{{{k^{\frac{1}{2}}}}}}$ and ${S_2} = \sum\limits_{k = 1}^n {\frac{1}{{{k^{\frac{1}{3}}}}}}$. How find all $n\in\mathbb{N}$ such that ${S_1} < ...
-1
votes
2answers
35 views

Understanding sub-sequences

Well, i currently studying about sequence and sub sequence and i noticed that i have problem with the definition. ...
0
votes
0answers
32 views

If no elements of a sequence $a_n$ are divisible by $\pi$, does $\forall n, a_n \mod \pi \in (0;\pi)$ hold?

Given a sequence like $a_n = n$ or $a_n = 50n$, (or any arbitrary constant), and that no element of the sequence is divisbile by $\pi$, would $b_n = a_n \mod \pi$ eventually take on all values in the ...
1
vote
0answers
55 views

Analytic continuation of the Dirichlet $\eta(s)$ series to $\Re(s) \gt -1$. Why does this work?

Take the known Dirichlet $\eta(s)$ series, $$\displaystyle \eta(s) = \sum _{n=1}^{\infty } \left( {\frac {1}{(2\,n-1)^{s}}} - \frac{1}{(2\,n)^s}\right), \qquad \Re(s)>0$$ and add $\displaystyle ...
0
votes
1answer
109 views

Simplify Infinite Series Involving Gamma Function $\Gamma$

This question originated from this problem. Can anyone help me simplifying the infinite series below: $$\sum_{n=0}^{+\infty}\frac{1}{\Gamma(\beta_2-n)}\frac{(-e^{-t})^n}{n!}$$ The only idea I have ...
1
vote
1answer
34 views

Find $c_0$ for which a sequence is in $l_2$?

Let $y \in l_2$ and let $$x_k=\left[C_0+\sum\limits_{j=0}^{k-1}\frac{y_j}{(\lambda+1)^{j+1}}\right](\lambda+1)^k.$$ Does there exits unique constant $C_0$, such that $x \in l_2?$ I need to show the ...
0
votes
0answers
23 views

Enumerating function of $a_n$ i.e. the function $\sum_{n=0}^{n=\infty}a_n x^n.$ [duplicate]

Consider the Fibonacci Series {$ a_n$} defined by $a_0=0,a_1=1,a_{n+1}=a_{n-1}+a_n $ for $n\ge1.$ Then what will be the enumerating function of $a_n$ i.e. the function ...
4
votes
2answers
101 views

Find the sum of the following infinite series

Find the sum of the following infinite series in which numerator and denominator contains term which are product of integers in arithmetic progression: $$\frac15+ ...
2
votes
5answers
156 views

How to prove $\lim_{n \to \infty}a_n=1 \rightarrow \lim_{n \to \infty}\sqrt[n] a_n=1$

Let $a_n\geq0$ Prove/disprove: $$\lim_{n \to \infty}a_n=1 \rightarrow \lim_{n \to \infty}\sqrt[n] a_n=1$$ Proof: By definition a sequence $\displaystyle\lim_{n \to \infty}\sqrt[n] b_n=L$ ...
4
votes
2answers
99 views

Is this inequality always valid? $\left|\sum_{i=0}^{\infty}x_i\right|\leq \sum_{i=0}^{\infty}|x_i|$

Let $x_i\in\mathbb{R}$ for all $i\in\mathbb{N}.$ Is the following inequality always true? $$\left|\sum_{i=0}^{\infty}x_i\right|\leq \sum_{i=0}^{\infty}|x_i|$$
1
vote
1answer
61 views

How to bound this sequence?

Consider $a_{n\:=\:}1\:+\sum _{k=1}^n\:\frac{2+k}{3^k+1}$. I want to show this sequence convrege using the Cauchy-theorem. So far this is what i wrote: Let $\epsilon \:>\:0$. we need to find that ...
2
votes
1answer
59 views

The stuttering sequences

Let's define a stuttering sequence the following way : Let $q\in\mathbb{N}^*,E_q=\{1,2,\dots,q\}$ and $(u_n)\in (E_q)^\mathbb{N}$. $(u_n)$ is a stuttering sequence of order $k$ with spacing $w$ iff ...
1
vote
4answers
206 views

Is this sequence theorem true?

If a sequence $\{a_n\}$ of non-negative reals is convergent, then $\{\sqrt a_n \}$ is also convergent. Is this proposition true? I think it is true but I don't know why it does make sense. If ...
2
votes
4answers
407 views

Prove that $\frac{\sin n}{n}$ is a Cauchy sequence from the definition.

Prove that $\frac{\sin n}{n}$ is a Cauchy sequence from the definition. The following is what I have tried: Suppose $n>m$ , $$|s_n-s_m|=\frac{\sin n}{n}-\frac{\sin ...
2
votes
2answers
105 views

Find $\frac{1}{\log 2}+\frac{1}{(\log 2)(\log 3)}+\frac{1}{(\log 2)(\log 3)(\log 4)}+ \cdots$

Is it possible to calculate the sum of $\dfrac{1}{\log 2}+\dfrac{1}{(\log 2)(\log 3)}+\dfrac{1}{(\log 2)(\log 3)(\log 4)}+ \cdots$?
1
vote
2answers
787 views

How to use a difference table to find a formula for a given sequence?

Given a sequence, how do you get a "rule" from its difference table? For example; 1, 2, 3, 4, 5, 6 1, 1, 1, 1, 1 Or 1, 2, 7, 16, 29 1, 5, 9, 14 4, 4, 4 I haven't really explored this ...
10
votes
2answers
225 views

Infinite series for the arctangent from the tangent of half-angle formula

From Hodge's biography of Turing: He had found the infinite series for the "inverse tangent function", starting from the trigonometrical formula for $\tan\left(\frac{1}{2}x\right)$.* The ...
0
votes
1answer
26 views

The longest increasing subsequence of a reversed sequence and a negated sequence

Let's say you have a sequence $A$, for example $1, 5, 2, 3, 6$. You take the reversed sequence: $6, 3, 2, 5, 1$ and the negated sequence: $-1, -5, -2, -3, -6$ and find the length of the longest ...
1
vote
3answers
682 views

Sum of a series of a number raised to incrementing powers

How would I estimate the sum of a series of numbers like this: $2^0+2^1+2^2+2^3+\cdots+2^n$. What math course deals with this sort of calculation? Thanks much!
4
votes
1answer
77 views

Convergence of power series with eventually constant coeffcients

Assume I have a sequence $f_n$ of power series of the form $$ f_n(x) = \sum_{i=0}^\infty{a_{n,i}x^i},\quad a_{n,i}=\begin{cases}\alpha_{n,i} & n\leq i,\\b_i & n>i.\end{cases}.\tag{*} $$ ...
0
votes
2answers
99 views

Given a divergent series, find a smaller divergent one. [closed]

Let $u_n$ be a positive sequence such that $\sum u_n$ diverges. Find $(v_n)$ such that $v_n=o(u_n)$ and $\sum v_n$ diverges. This is a difficult problem I'm stuck with. Can someone give me ...
1
vote
2answers
84 views

$\sum_{n=1}^{\infty}\frac{1}{n^{1+\left|{\cos n}\right|}}$ - convergent? [duplicate]

Is $\sum_{n=1}^{\infty}\frac{1}{n^{1+\left|{\cos n}\right|}}$ convergent? Yes because $|\cos n|>0$ and $\frac{1}{n^ \alpha}$ is convergent for $\alpha>1$. Is this a good way?
1
vote
1answer
38 views

Expand and hence find (Series)

After trying some more questions on Series I'm coming across problems that are rather similar but can't quite grasp what the question is asking for. The question is as follows: Write the first ...
0
votes
1answer
110 views

Changing the order summation and limit and proving a double-sequence identity

As a part of a work of mine I wanna use this claim (which I hope is true), and don't know why I can: Assume I have for every $i\in \mathbb N$ a series $\{a_i^n\}_{n\in\mathbb N}\subset\mathbb R$ ...
2
votes
2answers
264 views

Question on series till 2009

Numbers 1, 2, 3 ……, 2009 are written in the natural order. Numbers in odd places are stricken off to obtain a new sequence. Numbers in odd places are stricken off from this sequence to obtain ...
2
votes
1answer
27 views

Clarification on how to prove polynomial representations exist for infinite series

With reference to this question, I would like a clarification of the comment given by @Ant (but someone else could answer instead). I basically have 2 questions: Is there any formal way to prove ...
0
votes
1answer
50 views

Prove divergence of an alternating sequence

Prove that the divergence of the following sequence. $$s_n=\frac{(-1)^nn}{2n-1}$$ The following is the sample answer Note that $\exists N\in\mathbb{N}\, s.t.\,\forall k\geq N$ ...
0
votes
1answer
44 views

hard question on singularities

If every series converging to the singularity has a sub sequence such that limit of the function of the subsequence is zero what can the singularity be?
3
votes
2answers
50 views

The convergence of $ \sum_{n=2}^{+\infty} \frac{1}{\sqrt{n}} \ln(\frac{n+1}{n-1})$

Good morning, I have tried to prove the convergence with the application of the criterion of comparison. I have used this increase: $$ \sum_{n=2}^{+\infty} \frac{1}{\sqrt{n}} ...
0
votes
2answers
29 views

How to use the comparison test to show that this series diverges?

I have the following series: $\displaystyle\sum_{m=2}^\infty \left( \displaystyle\frac{5}{7 \, m + 28} \right)$ The partial sums are obviously smaller than the harmonic series, but that doesn't allow ...
2
votes
1answer
68 views

General Solution to functional equation

I was wondering how to derive the solution to $$ \frac{f(x + (1-2x)) - f(x)}{1-2x} = f(x)$$ Which can be simplified to $$\frac{f(1-x) - f(x)}{1-2x} = f(x)$$ One idea is as follows. Consider the ...
21
votes
2answers
992 views

What is wrong with the sum of these two series?

Could anyone help me to find the mistake in the following problem? Based on the formula of the sum of a geometric series: \begin{equation} 1 + x + x^{2} + \cdots + x^{n} + \cdots = \frac{1}{1 - x} ...
4
votes
0answers
103 views

An infinite series that gives $f(s)=s$. How could it be explained more easily?

This question loosely builds this one. Equate the following two infinite series for $\zeta(s)$: $$\displaystyle \zeta(s) = \frac{1}{4\,(s-1)} \left(1+s+\sum _{n=1}^{\infty } \left( {\frac ...
9
votes
1answer
341 views

Applications of equidistribution

What are applications of equidistributed sequences ? I'm looking for examples of problems/questions in fields (not necessarily mathematical) where equidistribution is an ad-hoc tool towards a ...
0
votes
3answers
49 views

Sum of series by comparing to a known Series expansion

I can't seem to quite grasp Summation of Series any bit and would like to ask for your help. The problem requires me to find the sum of series by using known series expansions. I have given the ...
8
votes
3answers
160 views

How can prove that $\sum_{n=1}^{\infty }\frac{\zeta (2n)}{4^{n-1}}(1-\frac{1}{4^n})=\frac{\pi }{2}$

$$\zeta (2)(1-\frac{1}{4})+\frac{\zeta (4)}{4}(1-\frac{1}{4^2})+\frac{\zeta (6)}{4^2}(1-\frac{1}{4^3})+...=\frac{\pi }{2}$$ The WolframAlph couldn't recognize the closed-form which is $\pi/2$ ...
1
vote
3answers
47 views

Mean of a vector

Be a set of numbers $v=(a_1, a_2, \ldots, a_n)$ I want to form the following average vector $\mu = (\frac{\sum a_i}{n}, \frac{\sum a_i}{n}, \ldots, \frac{\sum a_i}{n})$. If I do it iteratively step ...
3
votes
4answers
968 views

Prove that limit inferior is same as limit superior for a convergent sequence

I was reading the book "Understanding Analysis" by Stephen Abbott on my own. I came across the following problem. Let $(a_n)$ be a convergent sequence. Let $y_n$=sup{$a_k:k\geq n$}. Then lim sup ...