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

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0
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2answers
29 views

Find power series representation of $ x/(x^{2}+9)^{2}$

I'm not sure how to do it since the entire bottom term is squared. Is there a geometric series I should use? Or differentiation?
0
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2answers
20 views

Prove for a monotone function, if $x_0$ is interior on interval $I$ then the one-sided limits exists

The proof goes like this: Lets suppose $f$ is nondecreasing (for nonincreasing we'll observe the function $-f(x)$). Let $x_0$ an interior point in the interval $I$, and $\left\{x_k\right\}$ an ...
5
votes
3answers
155 views

Examples of “difficult” integrals with are easier to solve with a series?

Yesterday someone posted an extremely elegant solution to a seemingly bizarre series where the integral: $$\int_{0}^{1} x^{m}\ dx = \frac{1}{m + 1}$$ was utilized. Oftentimes one will also ...
4
votes
1answer
97 views

Is the sequences$\{S_n\}$ convergent? [duplicate]

Let $$S_n=e^{-n}\sum_{k=0}^n\frac{n^k}{k!}$$ Is the sequences$\{S_n\}$ convergent? The following is my answer,but this is not correct. please give some hints. For all $x\in\mathbb{R}$, ...
1
vote
1answer
47 views

sum of the series $\sum_{k=0}^{\infty}(k+1)(1-|x_k|)|x_n|^k$ [closed]

Let $|x_n|<1$. Is the given series convergent or which conditions do we impose on the sequence $(x_n)$ to given series be convergent? $\sum_{k=0}^{\infty}(k+1)(1-|x_k|)|x_n|^k$
1
vote
2answers
72 views

Find $R$ such that $\sum\limits_{n=2k}^{3k}\binom{3k}{n}\cdot{(R)}^n\cdot{(1-R)}^{3k-n}$ is constant for all $k\in\mathbb{N}$

Given $A_k = \sum\limits_{n=2k}^{3k}\binom{3k}{n}\cdot{(R)}^n\cdot{(1-R)}^{3k-n}$ with $0<R<1$. The sequence $A_k$ seems to be decreasing for $R\leq0.6$ and increasing for $R\geq0.8$. How can ...
1
vote
1answer
19 views

Is $(\frac{1}{1-|x_n|})$ bounded or only bounded above if $|x_n|<1$? [closed]

Let $(x_n)$ be a sequence such that $|x_n|<1$. Is $(\frac{1}{1-|x_n|})$ bounded or only bounded above if $|x_n|<1$?
1
vote
1answer
48 views

Is $(|x_n|^{n+1})$ convergent or bounded if $|x_n|<1$?

Let $(x_n)$ be a sequence such that $|x_n|<1$. Is $(|x_n|^{n+1})$ convergent or bounded?
3
votes
2answers
82 views

Prove by induction that $A_k = \sum\limits_{n=2k}^{3k}\binom{3k}{n}\cdot{(\frac{60}{100})}^n\cdot{(\frac{40}{100})}^{3k-n}$ is decreasing

I want to prove that the following sequence is monotonously decreasing: $A_k = \sum\limits_{n=2k}^{3k}\binom{3k}{n}\cdot{(\frac{60}{100})}^n\cdot{(\frac{40}{100})}^{3k-n}$ I think it should be ...
6
votes
2answers
243 views

A series with only rational terms for $\ln \ln 2$

We all know that $$ \ln 2 = \sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n}. $$ Do you know a series with only rational terms for $$\ln \ln 2 = ?$$ Let's exclude base expansions with non ...
12
votes
6answers
2k views

A strange puzzle having two possible solutions

A friend of mine asked me the following question: Suppose you have a basket in which there is a coin. The coin is marked with the number one. At noon less one minute, someone takes the coin ...
3
votes
3answers
99 views

Trouble evaluating the sum involving logarithm

I was trying to solve this problem: Closed form for $\int_0^1\log\log\left(\frac{1}{x}+\sqrt{\frac{1}{x^2}-1}\right)\mathrm dx$ In the procedure I followed, I came across the following sum: ...
3
votes
3answers
72 views

Closed form for the sum: $\sum_{n=1}^{\infty}\frac{1}{n(n + 1/3)}$ [duplicate]

I tried using partial fractions to compute the sum of the series $$ \sum_{n=1}^{\infty}\frac{1}{n(n + 1/3)} $$ Another technique is to turn this series into a definite integral of 0 to 1. but do not ...
7
votes
3answers
297 views

Sum related to zeta function

I was trying to evaluate the following sum: $$\sum_{k=0}^{\infty} \frac{1}{(3k+1)^3}$$ W|A gives a nice closed form but I have zero idea about the steps involved to evaluate the sum. How to approach ...
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
47 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 ...
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
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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 $$ ...
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)? ...
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 ...
6
votes
3answers
315 views

Sum of series with triangular numbers

Can you please tell me the sum of the seires $ \frac {1}{10} + \frac {3}{100} + \frac {6}{1000} + \frac {10}{10000} + \frac {15}{100000} + \cdots $ where the numerator is the series of triangular ...
2
votes
1answer
39 views

Generating a nonrandom sequence which has a normal distributed density

I need to create an algorithm in a computer program (Fortran90) which generates a sequence of $n$ (between $10$ and $10^6$) numbers $z$ that follow a normal distribution. Restrictions: Has to ...
5
votes
4answers
105 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
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 ...
0
votes
0answers
34 views

Proof of the sum of an alternating series similar to the alternating harmonic series [closed]

Hi I know I saw the answer to the following question somewhere online, I think it was stackexchange but I'm not absolutely certain..repeated searching on this website and on google have been ...
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
97 views

Is there some fast and efficient way for solving $x$

Let $b$ be a given constant scalar between 0 and 1, and $A$ a given $N \times N$ transitional probability matrix (i.e., each row sum of $A$ is 1, and $0\le {A}_{(i,j)} \le 1$). Let $A\circ A$ denote ...
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 ...
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 ...
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!
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
3answers
54 views

Geometric series of matrices

I am currently reading 'Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach' by J. Hubbard and B. Hubbard. In the first chapter, there is the proposition: Let A be a ...
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 ...
4
votes
2answers
45 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 ...
3
votes
1answer
53 views

Find a sequence

Find the function for the sequence $a_0 = 0, a_1 = 1$ and $a_{n}=a_{n+10}+a_n$ for all $n>0$.
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, $$ (E^2 + 1)u_n = \sqrt{2}\cos[(n-1)\pi/4] \quad \Rightarrow \quad u_n = \dfrac{1}{E^2 + 1}\{\sqrt{2}\cos[(n-1)\pi/4]\} $$ ...
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 ...
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 ...
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 ...
1
vote
0answers
25 views

Upper hemicontinuity and closed graphs

I have a problem with the definitions of upper hemicontinuity. In particular I found a picture that makes me wonder, not only about my understanding of this concept, but of the concept and application ...
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 ...
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?
3
votes
1answer
125 views

How to prove that $\sum_{k=0}^\infty \binom{x}{x-k}\cdot\binom{x}{k-x} = 1$?

How to prove this: $$\sum_{k=0}^\infty \binom{x}{x-k}\cdot\binom{x}{k-x} = 1$$ For all $x\in\mathbb R_{\ge0}$ and with $\binom{x}{r}=\frac{\Gamma(x+1)}{\Gamma(r+1)\cdot\Gamma(x-r+1)}$ It is obviously ...
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 ...
1
vote
3answers
83 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) = ...