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

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-1
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2answers
304 views

Mathematical continuation of sequence:$\space \ldots, 7,6,6,6,6,6,5,5,5,5,4,4,4,3,3,2,$?

Is it possible to mathematically deduce the next element (to the right) in the following series? It continues in the same pattern to the left ($n-1$ copies of the positive integer $n$ on the left). $$ ...
11
votes
3answers
594 views

Prove the Fibonacci sum $\sum \limits_{n=0}^{\infty}\frac{F_n}{p^n} = \frac{p}{p^2-p-1}$

We are familiar with the nifty fact that given the Fibonacci series $F_n = 0, 1, 1, 2, 3, 5, 8,\dots$ then $0.0112358\dots\approx 1/89$. In fact, $$\sum_{n=0}^{\infty}\frac{F_n}{10^n} = ...
2
votes
3answers
298 views

How to find $n$'th term of the sequence $3, 7, 12, 18, 25, \ldots$?

$$3, 7, 12, 18, 25, \ldots$$ This sequence appears in my son's math homework. The question is to find the $n$'th term. What is the formula and how do you derive it?
6
votes
1answer
104 views

Weaker assumption to ensure $f_{n}^\prime \to f'$

I learned in my real analysis class that if $f_n:[a,b] \to \mathbb{R}$ is a sequence of differentiable functions such that $f_n \to f$ uniformly and $f_{n}^\prime \to g$ uniformly then $f$ is ...
3
votes
3answers
750 views

How can I prove that the binomial coefficient ${n \choose k}$ is monotonically nondecreasing for $n \ge k$?

I want to prove that the binomial coefficient ${n \choose k}$ for $n \ge k$ is a monotonically nondecreasing sequence for a fixed $k$. How do I do this?
4
votes
4answers
192 views

What are some “natural” interpolations of the sequence $\small 0,1,1+2a,1+2a+3a^2,1+2a+3a^2+4a^3,\ldots $?

(This is a spin-off of a recent question here) In fiddling with the answer to that question I came to the set of sequences $\qquad \small \begin{array} {llll} ...
4
votes
1answer
311 views

tough sum/product series involving reciprocal of ln

I ran across an interesting series. I looked at it and must admit, I do not even know where to begin. I tried playing around with it, but to no avail. Here it is. Perhaps it isn't doable. ...
0
votes
1answer
141 views

Convergence of limit with integer part of $ x $

Good day! I tried to solve this problem;the process is correct? The problem si: Let $x\in\mathbb{R}$. With $[x]$ denote the integer part of $ x $. Calculate $$\lim_{x\to 0^+} \Biggr(x^2 ...
3
votes
3answers
505 views

How to get closed form from generating function?

I have this generating function: $$\frac{1}{2}\, \left( {\frac {1}{\sqrt {1-4\,z}}}-1 \right) \left( \,{ \frac {1-\sqrt {1-4\,z}}{2z}}-1 \right)$$ and I know that $\frac {1}{\sqrt {1-4\,z}}$ is ...
1
vote
1answer
145 views

Find a function for which the sequence converges uniformly

I'm stuck on a homework problem For each natural number $n$ and each number $x \in [0,1]$ let $$f_n(x) = \frac{x}{nx+1}$$ Find the function $f:[0,1] \to \mathbb{R}$ to which the sequence $\{f_n : ...
2
votes
2answers
186 views

Series Convergence for a variable

Fix a positive number $\alpha$ and consider the series $$\sum_{k=1}^\infty \frac1{(k+1)[\ln(k+1)]^\alpha} $$ For what values of $\alpha$ does this series converge? I can plug in a bunch of values, ...
2
votes
1answer
109 views

Convergence of a characteristic function

This is the last part of a three part problem on characteristic functions, and it's been driving me crazy over the last few days. Any help would be most appreciated. $X_1,X_2, \ldots, X_n$ are ...
0
votes
1answer
281 views

Reverse formula for arithmetic progression

Need formula for: f(x) => y where x and y are: 0..4 => 1 5..14 => 2 15..29 => 3 30..49 => 4 etc. i also need the opposite: f(a) => b,c where a,b and c are: 1 => 0,5 2 => 5,10 3 => 15,15 ...
9
votes
2answers
238 views

Erdős: Sum of rational function of positive integers is either rational or transcendental

I am trying to find a conjecture apparently made by Erdős and Straus. I say apparently because I have had so much trouble finding anything information about it that I'm beginning to doubt its ...
4
votes
4answers
540 views

Does every sequence have at least one limit point?

Let $x_n$ be a sequence of real numbers. Definition: $x \in \mathbb R \cup \{-\infty,\infty\}$ is a limit point of a sequence $x_n$ if there is a subsequence $x_{n_k}$ of our sequence such that ...
1
vote
1answer
163 views

Number of perturbations of the Jordan form

I am looking for information about the number of Jordan forms that can be obtained from a given Jordan form of a small perturbation. For example, if a Jordan form consists of a single cell $2 \times ...
4
votes
2answers
80 views

Sequences whose differences tend to $0$

Suppose $f(1,i)>0$ is a strictly decreasing sequence of reals. Let $f(k+1,n)=f(k,n+1)−f(k,n)$. If $f(2m+1,n)$ is for all integers $m$, a strictly decreasing function in $n$ and ...
1
vote
1answer
50 views

Convergence of $\infty$-accelerating rationals

Suppose $f(1,i)$ is a sequence of rationals above 1. Let $f(k+1,n)=f(k,n+1)-f(k,n)$. If $f(k,n)$ is for all $k>0$, an increasing function in n, must $\sum_{n>1} 1/f(1,n)<\infty$?
2
votes
0answers
97 views

About how big is $n^0+(n-1)^1+\cdots+0^n$? [duplicate]

Possible Duplicate: Asymptotics of $1^n + 2^{n-1} + 3^{n-2} +\cdots + (n-1)^2 + n^1$ About how big is is the sum $\sum_{k=0}^n k^{n-k}$? At the least, can we get an upper bound on it that ...
0
votes
1answer
71 views

How to prove this zeta function?

Prove that $\sum_{n=2}^{\infty} \frac{z^{n-1}}{\alpha(n-1)+1}$ is equivalent to $\frac{1}{\alpha} \displaystyle \int_{0}^{1}{ \frac{z t^{\frac{1}{\alpha}}}{1-tz}} dt$?
0
votes
2answers
644 views

Proving $\sum\limits_{i=0}^n i 2^{i-1} = (n+1) 2^n - 1$ by induction

I'm trying to apply an induction proof to show that $((n+1) 2^n - 1 $ is the sum of $(i 2^{i-1})$ from $0$ to $n$. the base case: L.H.S = R.H.S we assume that $(k+1) 2^k - 1 $ is true. we need to ...
2
votes
2answers
81 views

Determine the set of real numbers x for which the following series diverges

Determine the set of real numbers x for which the following series diverges $$\sum_{n=1}^{\infty}\left(\frac{1}{n}\csc\frac{1}{n}-1\right)^x$$
2
votes
1answer
92 views

Is the function $\sum_{n=0}^{\infty} \frac{n^2x}{1+n^4x^2}$ continuous in $x \in (0, 1]$?

We know that the function $\sum_{n=0}^{\infty} \frac{n^2x}{1+n^4x^2}$ is convergent but not uniformly convergent when $x$ is in $(0, 1]$. How do we know if it's continuous or not, though? Help much ...
1
vote
4answers
183 views

the sum of powers of $2$ between $2^0$ and $2^n$

Lately, I was wondering if there exists a closed expression for $2^0+2^1+\cdots+2^n$ for any $n$?
1
vote
1answer
386 views

How to calculate/approximate expectation of function of a binomial random variable?

I am stuck at following problem in my research. Suppose that $M=m$ is a random variable with binomial distribution and parameters $n,p$. The constants $r$ and $\gamma$ are greater than zero. ...
4
votes
1answer
72 views

Prove that $\sum_{n=1}^{\infty}x^n\frac{(n!)^3}{(3n)!}$ converges when $|x|$ < 27 and diverges when $|x| > 27$

This is a homework question that I am stuck on... I am not sure which test to use to prove this statement. If someone could let me know at least which test to use to push me in the right direction ...
0
votes
2answers
97 views

Why is the sequence divergent?

Well, the sequence is: $a_n = n! - n^n $ I can't seem to figure it out. $n!$ goes to infinity, $n^n$ goes to infinity, I know the the result should be negative infinity, but I can't really find a ...
2
votes
2answers
55 views

Alternating Sums of Two Sequences, One Termwise Dominating

Edited in light of Gerry Myerson's quick counterexample. I have two finite sequences $(a_n)$ and $(b_n)$ satisfying the following: all terms are positive both sequences are strictly decreasing the ...
6
votes
3answers
225 views

Sequence Convergence of $\sum_n\frac{(-1)^{n+1}}{3n + n(-1)^n}$

I have the following series $\displaystyle \sum_{n=1}^{+\infty} \frac{(-1)^{n+1}}{3n + n(-1)^n}$. Does it converge? I wanted to the alternating series test, but that's not easy because of the two ...
2
votes
2answers
104 views

Is the sequence $u_{n}=\sum_{p=0}^{n}(\arctan\frac{x}{2^p})^2$ convergent?

Let $x$ be a nonnegative real number. Is the sequence $\displaystyle \left\{\sum_{p=0}^{n}\left(\arctan\frac{x}{2^p}\right)^2\right\}_{n\geq 0}$ convergent ? It's easy to check that it is strictly ...
0
votes
2answers
199 views

Sum of reciprocals of powers [duplicate]

Possible Duplicate: “Closed” form for $\sum \frac{1}{n^n}$ Using Weierstrass theorem (any monotonic and bounded sequence is convergent), we can prove that the sequence ...
1
vote
3answers
560 views

Constructing a convergent subsequence

Let $a_{mn}$ be a double sequence in $[0,1]$. I would like to know whether I can do the following operation. I start with a sequence $a_{m1}$ and construct a convergent subsequence $a_{m'1}$. ...
9
votes
1answer
326 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 ...
1
vote
3answers
192 views

Summing a function using modulus

The problem: If the infinite sum of a function is known, how to find: $$\begin{align*} \sum_{i\equiv 0 \mod m}f(x_0+i)=\\ f(x_0)+f(x_0+m)+f(x_0+2m)+f(x_0+3m)+\ldots \end{align*}$$ And if the ...
0
votes
1answer
90 views

Singular point of a series in $\mathbb{C}$

I'm given the following problem. Let $f(x)=\sum c_nz^n$ in the unit disk, and furthermore $c_n\in \mathbb{R}, \ c_n >0$. Then the point $z_0=1$ is singular. I don't seem to understand this, ...
2
votes
2answers
151 views

Two Taylor expansions of $\frac{1}{1+\sqrt{2-z}}$ about $z=0$

How do you start expanding this function $$f(z)= \frac{1}{1+\sqrt{2-z}}$$ into two Taylor expansions about $z=0$? The best I came up is to let $u=\sqrt{2-z}$ and then expand $f(z)$ as a ...
1
vote
2answers
694 views

How to know if it diverges or converges and finding the convergent value

I am given the following succession/series/sequence: $$ a_n = \frac{4n^5 +4n^3+n}{5n^4-2n^5+n^2} $$ How do I find out if it converges or diverges and how to find such values. I am quite lost on the ...
1
vote
0answers
70 views

A composition on finite integral sequence

I want to know if there is a polynomial formula for this (in general): Given $f(x)=\sum _{i=0}^n a_ix^i$ where $a_i, x^i , n \in \mathbb N^* $ . Given $f_1(n) = f(x)$, we define recursively ...
8
votes
3answers
437 views

Prove that $\sum\limits_{k=1}^n \frac{1}{k^2+3k+1}$ is bounded above by $\frac{13}{20}$

I want ask a question about a sum. The exercise is as follows: Prove the following inequality for every $n \geq 1$: $$\sum\limits_{k=1}^n \frac{1}{k^2+3k+1} \leq \frac{13}{20} .$$
5
votes
1answer
551 views

$b_n$ bounded, $\sum a_n$ converges absolutely, then $\sum a_nb_n$ also

a) Prove that if $\sum a_n$ converges absolutely and $b_n$ is a bounded sequence, then also $\sum a_nb_n$ converges absolutely. I wanted to use the comparison test to show it's true, but I think I ...
26
votes
3answers
1k views

Why is this series of square root of twos equal $\pi$?

Wikipedia claims this but only cites an offline proof: $$\lim_{n\to\infty} 2^n \sqrt{2-\sqrt{2+... \sqrt 2}} = \pi$$ for $n$ square roots and one minus sign. The formula is not the "usual" one, like ...
1
vote
0answers
65 views

explicit formula for a sum of sequence [duplicate]

Possible Duplicate: Partial sum of rows of Pascal's triangle Working on the Euler characteristic of some topological space, I was led to the following situation. Let $a$ be a given ...
2
votes
1answer
123 views

Infinite product of recursive sequence

Let $a_{n+1}=\sqrt {(a_n+a_{n-1})/2}$ and $a_0=a_1=2$, how to prove convergence of the product $a_0 a_1 a_2 a_3...a_\infty$, and possibly find its value?
5
votes
2answers
154 views

Changing divergent series to convergent by re-ordering denominators

Suppose $a_n$ is strictly decreasing and positive and $\sum_{n>1}a_n/n=\infty$, let $g:\mathbb N\to\mathbb N$ be a bijection between the positive integers, can we have ...
2
votes
3answers
204 views

What is the expression for this summation?

Known that $\sum_{n=0}^{\infty}{x^n}{z^n}=\frac{1}{1-xz}$. If we have $\sum_{n=0}^{\infty}\frac{{x^n}{z^n}}{n\beta + \alpha}$ where $\beta, \alpha $ are element of real numbers but not equal $0$. ...
11
votes
4answers
607 views

Proof about an infinite sum

Hello I have a pretty elementary question but I am a bit confused. I am trying to prove that $$\sum_{k=1}^\infty \frac1{k^2+3k+1} \ge \frac12$$ thanks, Thrasyvoulos
11
votes
1answer
136 views

Irrationality of Two Series

Show that if the integers $1<b_1<b_2<\cdots$ increase so rapidly that$$\frac{1}{b_{k+1}}+\frac{1}{b_{k+2}}+\cdots<\frac{1}{b_{k}-1}-\frac{1}{b_{k}},\quad k\geq 1,$$ then the number ...
7
votes
1answer
335 views

For bounded sequences, does convergence of the Abel means imply that for the Cesàro means?

See the title. This is true if the sequence is nonnegative; some Tauberian theorems which I was able to find give some more general sufficient conditions. I would like to know if this is true for ...
1
vote
1answer
146 views

Existence of the limit of a sequence?

I solved this limit problem by following this way, but I'm not exactly sure about .... can anyone help me and tell me if it is correct? the problem is: Let $k>1$. If it exists, calculate the ...
7
votes
2answers
182 views

Does the family of series have a limit?

For $r<1$ define $F(r)=\sum_{n\in\mathbb N}(-1)^nr^{2^n}$. Does $F$ have a limit as $r\nearrow 1$?