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

learn more… | top users | synonyms (5)

3
votes
1answer
141 views

General Term of a Sequence

What would be the best way in finding a general term $a_n, n>3$ for the recursive sequence? $$a_n = \dfrac{6a_{n-1}^2a_{n-3} -8 a_{n-1}a_{n-2}^2}{a_{n-2}a_{n-3}}$$ where $a_1 = 1 ; a_2 = 2 ...
0
votes
1answer
61 views

Solving progressions! Blackjack safe bet.

I came to the conclusion that if I bet $x$ amount of money in blackjack and lost the next should be $2x$ and so on till I win. In the end, I will win at least the amount I first bet. So I came up with ...
7
votes
2answers
70 views

Infinite sequence of $3$ numbres with nonrepeated parts.

I am thinking about this problem. Can we construct infinite sequence with $3$ numbers so that no repeated parts exist in it? There should not be subsequence with $2k$ numbers so that its left and ...
1
vote
1answer
90 views

what value series $\sum_{r=1}^\infty\frac{\lfloor rp \rfloor}{2^r}$ converges?

I have been wondering about series of $$S=\sum_{r=1}^\infty\frac{\lfloor rp \rfloor}{2^r}$$ where p is a constant positive real number and $\lfloor\cdot\rfloor$ is floor function. I know it converges ...
1
vote
4answers
65 views

The interval determined by absolute value inequality $|1-2x| < |1+x|$

I'm working on a series convergence problem and am stuck on this part: The series converges when $|1-2x| < |1+x|$. How can I proceed from here to pick the values of $x$ that satisfy this ...
0
votes
1answer
38 views

On why a set is measurable ( a step of the proof of the Lebesgue monotone convergence theorem in Rudin).

Let $\{f_n\}$ be a sequence of measurable functions on $X$, and suppose that, for every $x \in X$: $$0 \le f_1(x) \le f_2(x) \le \cdots \le \infty $$ and $f_n(x) \rightarrow f(x) $ as $n \rightarrow ...
4
votes
1answer
76 views

Summation of a multiple series involving Fibonacci numbers

Compute the sum $$\sum_{a_{2015} = 0}^{\infty} \sum_{a_{2014} = 0}^{a_{2015}} \sum_{a_{2013} = 0}^{a_{2014}} \cdots \sum_{a_{1} = 0}^{a_2} \sum_{k=0}^{a_1} \frac{F_{k}}{2^{a_{2015}}} $$ where $F_k$ ...
2
votes
2answers
94 views

Summation of the telescoping series $\sum_{n=1}^N \frac{x}{(1+(n-1)x)(1+nx)}$

$(i)$ Verify that $$\frac{1}{1+(n-1)x} - \frac{1}{1+nx} = \frac{x}{(1+(n-1)x)(1+nx)}$$ $(ii)$ Hence show that for $x \ne 0$, $$\sum_{n=1}^N \frac{x}{(1+(n-1)x)(1+nx)}=\frac{N}{1+Nx}$$ Deduce that the ...
0
votes
2answers
97 views

The usage of notation $\lim a_n = 1^+$ for “$a_n$ approaches $1$ from above”

I have a seemingly trivial problem that I cannot seem to figure out. Imagine I want to prove that the series of $n$ does not converge, and for that I use the d'Alembert rule (yes it's ridiculous ...
0
votes
6answers
90 views

How to show this sequence converges to 1?

Let $x\:>\:1$ and consider $a_n$ a sequence of positive numbers and $\lim _{n\to \infty }\left(a_n\right)\:=\:0$ prove that: $$\lim _{n\to \infty }\left(x^{a_n}\right)\:=\:1$$ I thought about ...
2
votes
1answer
33 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
86 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
56 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
58 views

Sum of serie of integer that are halving

I'm trying to calculate the sum of integers that are halving (when the number is odd, we round down). Here are some example: S(100) = 100 + 50 + 25 + 12 + 6 + 3 + 1 + 0 + 0 + 0 + 0 + 0... S(3) = 3 ...
25
votes
1answer
421 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
104 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
117 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
57 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
42 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 ...
3
votes
2answers
94 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
80 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
49 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
30 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
28 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
48 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
57 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
30 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
22 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 ...
3
votes
2answers
74 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+ ...
4
votes
2answers
90 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
57 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
51 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
204 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
205 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
97 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
139 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
139 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
18 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
184 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
64 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
91 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
76 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
33 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
0answers
28 views

Operator of Taylor series as a distribution?

I would like to prove a statement: $$T:=\sum_{k=0}^\infty a_k\partial_x^k\delta_0\not\in\mathscr{S}'(\mathbb{R}^n)$$ and, in contrast, $$T_n=\sum_{k=0}^n ...
0
votes
1answer
91 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
223 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
21 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
35 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
41 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?