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

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2
votes
2answers
38 views

how to find convergence and divergence of the series [closed]

consider the following two series of complex numbers $$s_1=\sum_1^\infty\frac{i^{n}(2-\sin n)}{2^n.n}$$ $$s_2=\sum_1^\infty\frac{i^n(2-\sin n)}{2^n.n^2}$$ then find whether the above series ...
0
votes
3answers
48 views

How can I define a “formula” for general term of a sequence with some given values?

I have a doubt: If I have $\alpha, \beta, \gamma, \delta$ natural numbers, how can I write a formula to generate infinite sequences, such that $f(1)=\alpha, f(2)=\beta, f(3)=\gamma, f(4)=\delta$? I ...
-2
votes
3answers
169 views

How many is 1+2+3+… [duplicate]

Let $$S=1+2+3+\cdots=\sum_{n=1}^{\infty}n$$ What is the value of $S$? Some books says that $S=\infty$, other says that $S=-\frac{1}{12}$ and there are some books saying that this is a divergent ...
1
vote
2answers
68 views

If a series converges then the power series converges for all z

How can I prove that if $\sum \limits_{n=1}^{\infty} c_n$ , $c_n\in \mathbb{C}$, converges then $\sum \limits_{n=1}^{\infty} c_n \frac{z^n}{1-z^n}$ converges for all z in $\mathbb{C}$ with ...
1
vote
3answers
64 views

Calculating $\lim_{n\to\infty}\frac{\sum\limits_{i=0}^{n}{1.5}^i}{\sum\limits_{i=0}^{n}1}$

Given the sequence $\displaystyle{S_n}=\frac{\sum\limits_{i=0}^{n}{1.5}^i}{\sum\limits_{i=0}^{n}1}$ How can I calculate $\displaystyle\lim_{n\to\infty}S_n$ (or prove the divergence of this sequence)? ...
0
votes
1answer
43 views

How to prove this inequality by induction?

Suppose that $(v_n)$ is a sequence of positive real numbers with $v_1=1$ and such that $$ v_{n+1} \leq v_{n}+ \sqrt{v_{n}^2+1}. $$ How prove that $$ v_{n}\leq 2^n-1 $$ for any integer $n \geq 2$?
0
votes
0answers
37 views

Negation - Cauchy sequence

Suppose that $(x_i)_{i \geq 1}$ is not a Cauchy sequence of real numbers. How to prove that there exist $\varepsilon >0 $ and an increasing sequence $(i_n)$ of indices such that $$ ...
5
votes
4answers
107 views

Is $\sum\limits_{n=1}^\infty \sin{\frac{(-1)^{n+1}}{n}}$ convergent?

$$ \mbox{Is}\quad \sum_{n=1}^\infty \sin\left(\left[-1\right]^{n + 1} \over n\right) \quad\mbox{ convergent ?.} $$ $$ \mbox{I know that }\quad\sum_{n=1}^\infty \frac{(-1)^{n+1}}{n}\quad \mbox{is ...
1
vote
1answer
55 views

for which value of $a$ that$\sum_{n=0}^{\infty} \frac{1}{u_{n}^{a}}$ converges?

We are given an arbitrary real positive $u_0$. The sequence $\{u_n\}_{n\ge 0}$ is defined by $u_{n+1}=u_ne^{-u_n}$ for $n\ge 0$. Find the values of $a\in\mathbb{R}$ for which the sequence ...
4
votes
1answer
99 views

Finding a specific sequence of digits in pi

Looking at the pifs project on GitHub and this question on SO has made me curious as to how feasible it is to find a specific sequence of digits within Pi. Essentially, on average, how many digits of ...
0
votes
1answer
35 views

Sum of a sequence is smaller than 1

I try to understand a step in a proof. First, we define $\epsilon_n=1-\delta_0 - \delta_1-...-\delta_n$, where $\sum_j \delta_j =1$ and $\delta_j > 0 \ \forall j$. Claim: ...
3
votes
1answer
128 views

What is the 5000th happy prime number?

Im writing a program that finds the Nth happy prime number. I think it works, but to double check I want to compare what it returns for the 5000th happy prime number. The problem is, I dont know where ...
0
votes
1answer
65 views

Sum by twos for functions on $\Bbb{Z}$

I have two double sums with the steps $2$ and I do know that one of them is smaller than the other one (due to a complicated argument), but I would like to show it with direct computation. Let ...
0
votes
0answers
52 views

sum of the series $\sum_{k=0}^{\infty}(k+1)(x_n)^k.$

Let $x_n$ be a sequence of real numbers such that $x_n\in(0,1).$ Find the sum of the series $\sum_{k=0}^{\infty}(k+1)(x_n)^k.$ My answer is $\frac{(x_n)'}{(1-x_n)^2}.$ But the term $(x_n)'$ make me ...
2
votes
0answers
50 views

Find the limits of the convergent subsequences

Let my sequence be $a_n=n\pi-\lfloor n\pi\rfloor$ This sequence is bounded in $[0,1)$ so if must have a convergent subsequence. In fact, it seems to me like it has infinitely many convergent ...
7
votes
4answers
372 views

Evaluating $\sum\limits_{k=1}^{\infty}\frac1{(3k+1)(3k+2)}$ [duplicate]

What is the value of $\displaystyle\sum_{k=1}^{\infty}\frac1{(3k+1)(3k+2)}$?
0
votes
1answer
50 views

Help with finding a pattern

I'm currently working through a programming book and on one of the projects, I've hit a snag. I'm trying to avoid using a "magic number" solution and I am having trouble finding the relationship ...
0
votes
1answer
14 views

Adding Back Terms to solve for nth term

Q: If the first term of a sequence is $3$, and each of the following terms is found by multiplying the preceding term by $2$, what is the units digit of the $32$nd term in the sequence? My Attempt: I ...
1
vote
2answers
181 views

Solution to curious infinite series

How exactly does one find a closed form to: $$ \sum_{i=0}^{\infty}\left[\frac{1}{i!}\left(\frac{e^2 -1}{2} \right)^i \prod_{j=0}^{i}(x-2j) \right]$$ When expanded it takes on the form $$1 + ...
4
votes
2answers
59 views

what is the sum of this series: $\frac{2}{\pi}\Sigma_{k=1}^{\infty}\frac{(-1)^{k-1}}{2k-1}$

Can anyone help me with this? What is the sum of this series: $\frac{2}{\pi}\Sigma_{k=1}^{\infty}\frac{(-1)^{k-1}}{2k-1}$ I got it after plugging $x=-1$ in a fourier series Thank you!
2
votes
3answers
178 views

What is the closed form for $\sum_{n=1}^\infty \frac1n - \frac1{n+1/p}$?

A while ago, I started to look at expressions of the following form: $$ S_p:=\sum_{n=1}^\infty \frac1n - \frac1{n+1/p}, $$ where $p$ is prime, because otherwise things get too complicated for me at ...
4
votes
2answers
47 views

Upper bound of $\sum_{j=1}^p \frac{p+1}{p-j+1} \frac1{2^j}$

I am looking for an upper bound of the following sum $$ S_p:= \sum_{j=1}^p \frac{p+1}{p-j+1} \frac1{2^j}. $$ The upper bound should be independent of $p$, of course. Numerical experiments indicate ...
0
votes
3answers
278 views

What is the probability that A will win…

Two players are rolling two dices, if they get 6 Player A wins, if they get 7, player B wins, else they rolling the two dices again... What is the probability that A will win? I'd like to get any ...
1
vote
1answer
35 views

rearrangement of infinite sum

I would like to find a justification why it is correct to write for any non negative sequence $(a_{n,m})_{n,m} \subset \mathbb{R}$ that $$ \sum_{i=1}^\infty \sum_{j=1}^\infty a_{ij} = ...
2
votes
2answers
37 views

OEIS for Doubly Indexed Sequences

Is there an OEIS-like database for doubly indexed sequences? I feel like such a database would be extremely useful for mathematicians, and would be surprised if there wasn't one, but I can't seem to ...
9
votes
2answers
373 views

Derive a closed form for a sum with inverse binomial coefficients

First off, I would like to apologize again for the integral I posted several days ago involving $\zeta(5)$. I was careless and did not examine the decimals out far enough. With that said, I would ...
8
votes
2answers
306 views

Sum of squares of binomial coefficients

I came across the following sum in reference to this question $$\sum_{n=0}^{\infty} \frac{1}{2^{5 n}} \binom{2 n}{n}^2 = \frac{\sqrt{\pi}}{\Gamma \left( \frac{3}{4}\right)^2}$$ The sum on the left ...
1
vote
1answer
54 views

How to graph $\sum\limits_{k=-\infty}^{\infty} \frac{-2x}{\pi^2(2k+1)^2+x^2}$

How do I plot the following function by hand without using the aid of a computer? $$\sum\limits_{k=-\infty}^{\infty} \frac{-2x}{\pi^2(2k+1)^2+x^2}$$
2
votes
1answer
44 views

Particular solution of the recurrence equation $u_{n+2} + u_n = \sqrt{2}\cos[(n-1)\pi/4]$

I would like to solve the equation xx recurrence using the operator $E$, ie, $$ (E^2 + 1)u_n = \sqrt{2}\cos[(n-1)\pi/4] \quad \Rightarrow \quad u_n = \dfrac{1}{E^2 + 1}\{\sqrt{2}\cos[(n-1)\pi/4]\} $$ ...
6
votes
3answers
96 views

Series involving log $\sum_{n=1}^{\infty} \left( n\log \left(\frac{2n+1}{2n-1}\right)-1\right) = \frac{1-\log 2}{2}$

Does anybody know how to prove this series? $$\sum_{n=1}^{\infty} \left( n\log \left(\frac{2n+1}{2n-1}\right)-1\right) = \frac{1-\log 2}{2}$$ I arrived at this through Mathematica. I tried writing ...
1
vote
1answer
47 views

Finding the sum of a finite arithmetic series where the common ratio changes

I realise how to find the sum up a finite arithmetic series when the common ratio is the same each time. 1/2n(2a+(n-1)d) However what happens when d (the common ...
1
vote
1answer
32 views

Rearrangements of Dirichlet Eta Function

I was wondering if explicit closed forms for rearrangements of $\eta(s)$, for $s$ such that the series is not absolutely convergent, are useful in studying the Dirichlet $\eta$ function. I am asking ...
2
votes
1answer
54 views

If a series converges, does it converge with additional log term multiplied?

If $\sum_{n} |a_n| < \infty$, is it true that $\sum_{n} |a_n\log(a_n)| < \infty$ if $0 \leq a_n \leq 1$? I am trying to see if $A$ is trace class operator, then $A \log(A)$ is also trace class ...
7
votes
2answers
158 views

Prove that $\displaystyle{\sum_{n=1}^{\infty}}(-1)^{n-1} \dfrac{H_n}{n} = \dfrac{\pi^2}{12} - \dfrac{1}{2}\ln^2 2$

We know that $H_n = \sum_{j=1}^{n}{1 \over j}$. Article in The Sum of Certain Series Related To Harmonic Numbers of Omran Kolba, we have proof of this identity which involves some advanced concepts. ...
1
vote
2answers
72 views

Show $ \sum_{n=1}^\infty { {\left(-1\right)^{n+1}} \over \Gamma(n+1)} = 1 - \frac1e $

Show $$ \sum_{n = 1}^{\infty}{\left(-1\right)^{n+1} \over \Gamma\left(n + 1\right)} =1 - {1 \over {\rm e}} $$ This may be related to $$e=\sum_{n=0}^\infty \frac1{n!}$$ I am having trouble ...
11
votes
4answers
1k views

How to solve this sequence $165,195,255,285,345,x$

This is a question appeared in a competitive exam. The question is: Find the unknown term in $165,195,255,285,345,x$ 1)375 $\ \ \ \ \ \ \ \ $ 2)420 3)435 $\ \ \ \ \ \ \ ...
1
vote
3answers
164 views

Series Convergence.

How does the series $\sum_{i=1}^\infty$ $\sqrt{2n+1}/n^2$ converge? I have yet to recieve a result that is not inconclusive. If you could tell me what test you used to confirm its convergence that ...
0
votes
0answers
12 views

How to prove a function $f(t)=\sum_{n=0}^{\infty} a_n t^{2n}$ with real coefficients $a_n$ is regular in the strip $-\pi/4\le \Im(t) \le \pi/4$?

How to prove a function $f(t)=\sum_{n=0}^{\infty} a_n t^{2n}$ with real coefficients $a_n$ is regular in the strip $-\pi/2\le \Im(t) \le \pi/2$? What will be the sufficient conditions on the real ...
5
votes
1answer
84 views

An inequality about a sequence

Let $(a_n)$ be a sequence such that $a_0=1 , a_1=2 , a_{n+1}=a_n+\dfrac {a_{n-1}}{1+ a_{n-1}^2} , \forall n \ge1 $ , then is it true that $52 < a_{1371} < 65$ ? $ EDIT:-$ I am posing another ...
1
vote
3answers
84 views

Particular solution of RE: $u_{n+1} - 2u_n = n^22^n$

Find the particular solution of recorrence equation $u_{n+1} - 2u_n = n^22^n$. I am developing a practical method using operators $E$ e $\Delta$, defined by $E(u_n) = u_{n+1}$ and $\Delta(u_n) = ...
2
votes
1answer
51 views

$a_n$ diverge $\nRightarrow a^2_n - a_n + 1$ diverges

Let $a_n$ be divergent sequence. Then a sequence $a^2_n - a_n + 1$ diverges. I have difficulties with finding out a counterexample. Could you help me?
1
vote
4answers
52 views

$\lim_{n \rightarrow \infty} f_n(x) = n^2 \left( 1- \cos \frac{x^3 - 1}{n} \right)$

Let $$f_n(x) = n^2 \left( 1- \cos \frac{x^3 - 1}{n} \right)$$ Let M be the set of x s.t. $\lim_{n \rightarrow \infty} f_n(x)$ exists. For each $x \in M$ let $f(x) = \lim_{n \rightarrow ...
2
votes
2answers
60 views

Does this series violate the decreasing condition of the Integral Test for Convergence?

I'm working on the section involving the Integral Test for Convergence in my calculus II class right now, and I've run into a seeming conflict between the definition of the Integral Test, and the ...
2
votes
2answers
36 views

convergence of a sequence

I'm reading a paper, for proving a claim it defines $$ f_n(x) = \dfrac{(rx-x^2)^n}{n!} $$ when $ r = \frac{a}{b} $ is a rational, and $ I_n = \int^r_0 f_n(x) \cdot \sin x \cdot dx $ , and then it says ...
2
votes
2answers
44 views

Determine if the alternating series converges absolutely, conditionally or diverges

Trying to determine if this alternating series converges absolutely or conditionally. ATS criteria has been met (terms are positive [ignoring signs] & decreasing, and the lim n->inf = 0, assuming ...
3
votes
0answers
40 views

Prove that $I = \int_0^{m(m+1} y_n(x)\,\mathrm{d}x$ converges and $I \in \mathbb{Q}$.

My problem is stated as follows Let $y_0(x) = x, \ \: y_1(x) = \sqrt{x}, \ \: y_{n+1}(x) = \sqrt{y_n(x) +x\,} \ $. Now define $ \displaystyle \hspace{3cm} I_n = \int_0^k ...
3
votes
2answers
33 views

What is the range of convergence of $\sum_{k=0}^\infty (k \cdot x \exp(-x))^k\cdot {1 \over k!} \cdot {1\over k+1}$?

I have the following series as an expression which occurs as a limit of a quotient of polynomials in $e$ and $x$ which I've expanded by polynomial long division into a series: $$f(x) = ...
1
vote
1answer
72 views

Expected value over many trials

I am a poker player and was talking to my friend about expected value. He claimed that if you play far enough above your bankroll, expected value can be negative, even if you have a skill edge. I ...
2
votes
1answer
47 views

Identity with binomials [duplicate]

Does there exist a closed formula for $$\underset{n=1}{\overset{N-1}{\sum}}\dbinom{N+n}{n}?$$ I've searching on wikipedia but I haven't found this kind of sum.
2
votes
1answer
300 views

Bonferroni's Principle

I am working with following Principle (My Approach At Bottom) : 1.2.3 An Example of Bonferroni’s Principle Suppose there are believed to be some “evil-doers” out there, and we want to detect them. ...