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

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2
votes
1answer
56 views

Looking for formula of $\sum_{k=1}^m (-1)^k \dfrac {x^2(x^2-1)…(x^2-k+1)}{(x+1)(x+2)…(x+k)}$

Let \begin{equation*} u_k:=(-1)^k \dfrac {x^2(x^2-1)...(x^2-k+1)}{(x+1)(x+2)...(x+k)}. \end{equation*} Can we find the sum of first $m$ of $u_k$ 's? That is, is there any formula for $\sum _{k=1}^m ...
26
votes
4answers
3k views

Showing that $\frac{\sqrt[n]{n!}}{n}$ $\rightarrow \frac{1}{e}$

Show:$$\lim_{n\to\infty}\frac{\sqrt[n]{n!}}{n}= \frac{1}{e}$$ So I can expand the numerator by geometric mean. Letting $C_{n}=\left(\ln(a_{1})+...+\ln(a_{n})\right)/n$. Let the numerator be called ...
0
votes
2answers
30 views

Recursive formula of an explicitly defined sequence

Does there exist a recursive formula for this sequence? $$a_n=\frac{2}{3}\left(1-\left(-\frac{1}{2}\right)^n\right), n\in\mathbb{N}_0$$
2
votes
1answer
49 views

Prove that all the five sequences converge to the same point $P \in \mathbb{R}^3$.

Let five sequences $A_n, B_n, C_n, D_n, E_n \in \mathbb{R}^3$ be constructed as follows: $A_0, B_0, C_0, D_0$ and $E_0$ are some given points of the space and $A_{n+1}, B_{n+1}, C_{n+1}, D_{n+1}, ...
3
votes
3answers
51 views

$\lim \limits_{n \to \infty}$ $\prod_{r=1}^{n} \cos(\frac{x}{2^r})$

$\lim \limits_{n \to \infty}$ $\prod_{r=1}^{n} \cos(\frac{x}{2^r})$ How do I simplify this limit? I tried multiplying dividing $\sin(\frac{x}{2^r})$ to use half angle formula but it doesnt give ...
5
votes
1answer
113 views

If $\sum (a_n)^2$ converges and $\sum (b_n)^2$ converges, does $\sum (a_n)(b_n)$ converge?

Could someone help me to solve this or at least give me a hint?, I have tried using Cauchy's criterion, the Dirichlet test for convergence, etc, but I can´t prove it.Honestly I don´t know where to ...
1
vote
1answer
37 views

Find new generating function, given an arbitrary generating function

In a discrete mathematics past paper, I am asked to find the generating function for the sequence $$\langle a_0, 0, a_2, 0, a_4, 0, \ldots \rangle,$$ given that $A(x)$ is the generating function for ...
3
votes
1answer
47 views

If $\sum_n \frac{1}{\alpha_n}$ is convergent, what can we say about $\min_{r,s}{}_{+} |\alpha_r-\alpha_s|$?

Let $\alpha_n$ a sequence of real numbers. If $$\sum_n \frac{1}{\alpha_n}$$ is convergent, what can we say about $\min_{r,s}{}_{+} |\alpha_r-\alpha_s|$? I thought, for example, you could say ...
0
votes
1answer
548 views

Prove that absolute convergence implies unconditional convergence

In the proof of "absolute convergence implies unconditional convergence" for a convergent series $\sum_{n=1}^{\infty}a_n$, we take a partial sum of first $n$ terms of both the original series ($S_n$) ...
5
votes
1answer
69 views

Cauchy-Ramanujan Formula $ \displaystyle \sum_{\stackrel{m \in \mathbb{Z}}{m \neq 0}} \frac{\coth m \pi}{m^{4p+3}} $

Cauchy and Ramanujan both gave the formula: $$ \sum_{\stackrel{m \in \mathbb{Z}}{m \neq 0}} \frac{\coth m \pi}{m^{4p+3}} = (2\pi)^{4p+3}\sum_{k=0}^{2p+2} (-1)^{k+1} ...
0
votes
1answer
25 views

How to calculate the number of combinations of $x$ integers, each with a value between $y$ and $z$?

For example, if I have 4 integers, and each can be between 0 and 36, how many combinations are there? If the numbers have appeared before, but in a new order, then this still counts as a new ...
2
votes
2answers
35 views

Computing the value of a series by telescoping cancellations vs. infinite limit of partial sums

$$\sum_{m=5}^\infty \frac{3}{m^2+3m+2}$$ Given this problem my first approach was to take the limit of partial sums. To my surprise this didn't work. Many expletives later I realized it was a ...
4
votes
1answer
21 views

How to make a topology out of $N$ that involves convergent / divergent sets.

Let $N$ be the naturals $1, 2, \dots$ Call a subset $A$ of $N$ convergent if the reciprocal sum $\sum_{a \in A} \frac{1}{a}$ converges. Similarly call as set divergent if the sum diverges. Notice ...
1
vote
3answers
49 views

Explicit form of this series expansion?

I am considering the following series expansion: $$f(k):=\sum_{n\geq 1} e^{-k n^2}$$ with $k>0$ a fixed parameter. Is there a possibility to either find a closed form expression for $f(k)$? Or at ...
0
votes
1answer
34 views

About inequality $\sum_{k=1}^n |a_k|^2 \lessgtr \sum_{k\neq s} |a_k| |a_s|$

Let $a_k$ a sequence of complex number. We have $$\left(\sum_{k=1}^n |a_k|\right)^2 \geq \sum_{k=1}^n |a_k|^2$$ It is a basic fact because $$\left(\sum_{k=1}^n |a_k|\right)^2 = \sum_{k=1}^n |a_k|^2 + ...
-1
votes
0answers
32 views

Show that $x_n \to x$ in $(X,d_f)$ iff $f(x_n) \to f(x)$

Let $(x_n)$ be a sequence in $X$ and $x \in X$. $(X,d)$ is a metric space and $f: X \times X\to $ is a bijection. Show that $x_n \to x$ in $(X,d_f)$ iff $f(x_n) \to f(x)$. ...
1
vote
4answers
110 views

The convergence of the series $\sum (-1)^n \frac{n}{n+1}$ and the value of its sum

This sum seems convergent, but how to find its precise value? $$\sum\limits_{n=1}^{\infty}{(-1)^{n+1} \frac{n}{n+1}} = \frac{1}{2}-\frac{2}{3}+\frac{3}{4}-\frac{4}{5}+...=-0.3068... $$ Any help ...
4
votes
2answers
47 views

Triangular Array's Recursive Formula Breakdown

I have the following polynomials: $$1$$ $$z-1$$ $$z^2-2z+3$$ $$z^3-3z^2+9z-15$$ $$z^4-4z^3+18z^2-60z+93$$ $$z^5-5z^4+30z^3-150z^2+465z-725$$ $$...$$ They are generated both recursively and explicitly. ...
1
vote
1answer
165 views

If two sequences converge in a metric space, the sequence of the distances converges to the distance of limits of the sequences.

Let $(p_n)$ and $(q_n)$ be sequences in the metric space $(X, d)$ and assume that $p_n \rightarrow p \in X$ and $q_n \rightarrow q \in X$. Prove that $d(p_n, q_n)$ converges to $d(p, q)$. Ok, so ...
2
votes
1answer
34 views

Find an example that the following equality doesn't apply

I need a sequence $(f_n)_{n\in\mathbb{N}}, f_n:X\to\mathbb{R}^\star$ which is measurable, $f_n \ngeq 0$ and doens't accept the following equality: $$\int_X\sum_{n=1}^\infty f_n \, d\mu = ...
2
votes
1answer
27 views

Normal convergence: $\sum_{n = 1}^{\infty} \frac{x}{(1 + (n-1)x)(1 + nx)}$

I want to prove that: $$\sum_{n = 1}^{\infty} \frac{x}{(1 + (n-1)x)(1 + nx)}$$ is not normally convergent on $[a, \infty)$ for fixed $a>0$. Let $U_n(x)$ denote the general term. We have: ...
1
vote
1answer
11 views

Sequence of functions that extends the algebraic properties of exponents to higher level operators.

I was thinking about some simple algebraic exponent properties such as the following $$ z^{x+y} = z^xz^y $$ and I started wondering about analytically continuing this identity to "higher-level ...
2
votes
2answers
205 views

Easy question about finite energy due to convergence

The infinite-length sequence $x_1[n]$ defined by \begin{multline} x_1[n]= \begin{cases} \dfrac{1}{n}& \text{if $n \geq $1},\ 0& \text{if $n \leq $0}. \end{cases} \end{multline} has an energy ...
3
votes
0answers
41 views

Elemenatry topological proof of Erdos conjecture on Arithmetic Sequences

Define $C_n(A) = \{a \in A : \forall d \in \Bbb{N}$ one of $a + d, a +2d, \dots a + (n-1)d$ is not in $A\}$ For $n \leq m$, we have $C_n(A) \subset C_m(A)$. Proof. Let $x$ be in the LHS. Then ...
3
votes
0answers
43 views

Closed form of an infinite series of integrals $\int_{0}^{\eta} \cos nt \cos t \sqrt{\cos^2 t - \cos^2 \eta}$

Let $$ I(n,\eta) = \int_{0}^{\eta} \cos nt \, \cos t \, \sqrt{\cos^2 t - \cos^2 \eta}\; dt $$ where it is known that $0 < \eta \leq \frac \pi 2$. Is it possible to evaluate $S$, the infinite ...
2
votes
2answers
57 views

How to evaluate $\sum_{j=0}^\infty\;\sum_{\substack{k=0 \\ k \neq j}}^\infty \frac{1}{j^2-k^2}$

I was reading an introductory text on multiple integrals and I have encountered a problem asking me to explain why $$ \sum_{j=0}^\infty\;\sum_{\substack{k=0 \\ k \neq j}}^\infty ...
2
votes
1answer
64 views

Difficult sequence puzzle

What is the 25$^{th}$ term of this infinite sequence? $$1,1,1,1,1,691,2,3617,...$$ I have tried for an hour now and I can't find any meaningful relation between the terms.
-1
votes
1answer
60 views

How can I find out if a non-convergent series is “indeterminate” (that is, “oscillating”) or “divergent”?

Definitions: Given a sequence $\{a_n\}$, define $$s_n= \sum_{j=0}^n a_j.$$ The sequence $\{s_n\}$ is called the series of partial sums of $\{a_n\}$. A series is convergent if $\{s_n\}$ has ...
2
votes
3answers
43 views

How to prove that $\sum \sin(2\pi n! x)$ does not converge uniformly on $\mathbb Q$

I want to prove that: $$\sum_{n \ge 1} \sin(2 \pi n! x)$$ converges absolutely on $\mathbb Q$, but not uniformly. For absolute convergence, let $x \in \mathbb Q$, then we can write: $x = a/b$ for ...
28
votes
2answers
1k views

Find the exact value of the infinite sum $\sum_{n=1}^\infty \{\mathrm{e}-(1+\frac1n)^{n}\}$

How can we find the exact value of the infinite sum $$ \displaystyle\sum_{n=1}^\infty \left\{\mathrm{e}-\Big(1+\frac1n\Big)^n\right\}? $$ This problem appears in: T. Andreescu, T. Radulescu & ...
0
votes
1answer
194 views

Solving this set of quadratic equations

I have a set of quadratic equations of the form: \begin{equation*} 2S_0q_0(q_0S_0 + q_1S_1 + ...... + q_iS_n)+k_0 = 0 \\ 2S_1q_1(q_0S_0 + q_1S_1 + ...... + q_iS_n)+k_1 = 0 \\ \vdots \\ ...
5
votes
2answers
56 views

Find the generating function of this sequence

I need to find the generating function of the sequence $c_n = (a_0, a_1, a_2, \ldots)$, where: $$a_n = \begin{cases} 2^{n/2} & \text{if $n$ is even,} \\ 1 & \text{if $n$ is odd.} ...
0
votes
0answers
37 views

Computing an exponential generating function from the first few terms

The current question is related to this one, and this other one. I have a number sequence, and I want to find generating ...
4
votes
1answer
40 views

Variant Generating Function related to Euler Numbers

The generating function $$\frac{2e^x}{e^{2x}+1}=\sum_{n\ge 0}E_k\frac{x^k}{k!}$$ counts the number of alternating permutations of a set with an even number of elements. My question is this, if we ...
5
votes
1answer
97 views

Does this sum converge or not?

I have the following sum:$$1-\frac{1}{2}-\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}-\frac{1}{7}-\frac{1}{8}-\frac{1}{9}-\frac{1}{10}+++++------...$$ How can I get an expression for its partial ...
6
votes
2answers
138 views

Sum $\sum_{n=2}^{\infty} \dfrac{n^4+3n^2+10n+10}{2^n(n^4+4)}$

I want to evaluate the sum $$\large\sum_{n=2}^{\infty} \frac{n^4+3n^2+10n+10}{2^n(n^4+4)}.$$ I did partial fraction decomposition to get ...
3
votes
2answers
34 views

Total area for a natural nested set of convex polygons.

Suppose we have a convex polygon $P_0$ with $n$ given vertices, and we want to "nest" polygons $P_j$ for $j > 0$ by taking the midpoints between edges of $P_{j-1}$ as the vertices. For a regular ...
2
votes
1answer
72 views

Is it always true that no closed forms exists for any divergent series?

Having seen many questions regarding finding closed form of integrals or infinite series, and some users providing either the final answer or detailed solution, and also reading how one finds a closed ...
7
votes
3answers
236 views

Identity with Harmonic and Catalan numbers

Can anyone help me with this. Prove that $$2\log \left(\sum_{n=0}^{\infty}\binom{2n}{n}\frac{x^n}{n+1}\right)=\sum_{n=1}^{\infty}\binom{2n}{n}\left(H_{2n-1}-H_n\right)\frac{x^n}{n}$$ Where ...
1
vote
2answers
90 views

$\sum_{n=1}^\infty {a_n}$ converges $\iff \sum_{n=1}^\infty {a_{n_k}}$ converges.

Let $({a_n})_{n\in{\mathbb{N}}}$ a sequence,and let $({a_{n_k}})_{k\in{\mathbb{N}}}$ the sequence of all terms of $({a_n})$ different than zero. Then $\sum_{n=1}^\infty {a_n}$ converges iff ...
24
votes
3answers
1k views

Apery's proof of the irrationality of $\zeta(3)$

I am trying to understand Apery's 1978 proof that $\zeta(3) = \displaystyle \sum_{n=1}^\infty \frac{1}{n^3}$ is irrational. The idea behind the proof is to find an 'accelerated' series for $\zeta(3)$ ...
17
votes
2answers
1k views

Evaluating $\sum_{n=1}^{\infty} \frac{n}{e^{2 \pi n}-1}$ using the inverse Mellin transform

Inspired by this post, I'm trying to evaluate $\displaystyle \sum_{n=1}^{\infty} \frac{n}{e^{2 \pi n}-1} = \frac{1}{24} - \frac{1}{8 \pi}$ using the inverse Mellin transform. But my answer is twice ...
0
votes
1answer
21 views

Proof of Lagrange Identity

I need to prove Lagrange Identity for complex case, i.e. $$ \left( \sum_{i=1}^n|a_i|^2 \right)\left( \sum_{i=1}^n |b_i|^2 \right)-\left| \sum_{i=1}^na_ib_i \right|^2=\sum_{1\leq i<j\leq ...
2
votes
1answer
35 views

$\sum_{n=1}^\infty {a_n}$ is absolutely convergent and ${b_n}$ is any subsequence of ${a_n}$, then $\sum_{n=1}^\infty {b_n}$ is abs. convergent

If $\sum_{n=1}^\infty {a_n}$ is absolutely convergent and ${b_n}$ is any subsequence of ${a_n}$, then $\sum_{n=1}^\infty {b_n}$ is absolutely convergent. My attempt of proof: Let ${b_j}={a_{n_j}}$ ...
4
votes
2answers
180 views

Finite Summation of Fractional Factorial Series

Is there a closed form solution for the following series? (Without Using Gamma Function): $$ S=\sum _{i=1}^{n-1} \frac{1}{(i+1)!} $$
0
votes
2answers
32 views

Alternating Reciprocal of Squares

I know that the infinite sum of the reciprocals of squares converges to $\pi^2/6$. Interested by this, I looked at a different sum. It is similar to the previously mentioned series, but it alternates ...
2
votes
3answers
54 views

Convergence of $\sum_{n=0}^\infty \frac{\ln(1+2^n)}{n^2+x^{2n}}$

Let $x \in \mathbb{R}$. Define the series: $$\sum_{n=0}^\infty \frac{\ln(1+2^n)}{n^2+x^{2n}}.$$ For what $x$ does it converge? It clearly has positive terms. The ratio and root tests seem ...
4
votes
0answers
34 views

Questionable Convergence of a Series

The summation is: $$ S = \sum_{k \geq 0} f(k) \int_{0}^{\pi/2} \sqrt{1-(1- \frac{f(k+1)^2}{f(k)^2})\sin^2(\theta)}d\theta $$ Now, we know that $f(k+1) < f(k)$ and as $k$ approaches infinity, ...
0
votes
1answer
18 views

Applicability of tests of convergence for series with non-negative terms

We know that there are many criteria of convergence for series with non-negative terms (for example, ratio test (with limit), root test (with limit), integral, comparison, and asymptotic comparison, ...
8
votes
2answers
311 views

Compute $\sum\limits_{n = 1}^{\infty} \frac{1}{n^4}$.

Compute the Fourier series for $x^3$ and use it to compute the value of $\sum\limits_{n = 1}^{\infty} \frac{1}{n^4}$. I determined the coefficients of the Fourier series, which are $$a_0 = ...