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

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20
votes
2answers
230 views

How to prove $\displaystyle\sum_{n=0}^{\infty} \dfrac{1}{1+n^2} = \dfrac{\pi+1}{2}+\dfrac{\pi}{e^{2\pi}-1}$

How can we prove the following $$\sum_{n=0}^{\infty} \dfrac{1}{1+n^2} = \dfrac{\pi+1}{2}+\dfrac{\pi}{e^{2\pi}-1}$$ I tried using partial fraction and the famous result $$\sum_{n=0}^{\infty} ...
2
votes
1answer
35 views

Series does not converge [closed]

How would I go about showing that the series$$\sum_{n + m\tau \in \Lambda} {1\over{{|n + m\tau|}^2}}$$does not converge, where $\tau \in \mathbb{H}$?
-1
votes
4answers
40 views

The next number in series

What is the next number in the following series: 12, 35, 81, 173, 357 , ___ I am not able to find the answer. Please explain.
2
votes
2answers
101 views

Is $\sum_{n=1}^\infty \frac{1}{2^{n^2}}$ is rational number?

Can anyone help with this: Is $\sum_{n=1}^{\infty}\frac{1}{2^{n^2}}$ is rational number?
0
votes
0answers
14 views

Incrementing an answer by one by using a limited set of numbers

I have recently taken part in a counting experiment on reddit and wondered the following. Is there an equation or simplified (not necessarily simple) theory to calculate which functions to use to ...
1
vote
1answer
33 views

Problem on uniform convergence

$h : \Bbb R \times S \to \Bbb R$ continuous. $x_n \to x$. $S$ compact. Does $h(x_n,.)\to h(x,.)$ uniformly ? I know that pointwise convergence and equicontinuity implies uniform convergence on ...
2
votes
3answers
79 views

Help me find the following limit : $\lim_{{n}\to{\infty}} (\frac{2^x+3^x+\cdots+n^x}{n-1})^\frac{1}{x} = ?$

I have no idea where to start.$$\begin{align}\lim_{{n}\to{\infty}} \left(\dfrac{2^x+3^x+\cdots+n^x}{n-1}\right)^{1/x} = ?, n >1\\\end{align}$$
3
votes
3answers
81 views

Sum $\sum_{n=1}^{\infty}\ln(1+2^{-2^n})$

How can I solve this: Find the sum of: $$\sum_{n=0}^{\infty}\ln(1+2^{-2^n})=?$$ and $$\sum_{n=1}^{\infty}{(-1)^{n+1}}\cdot{1\over n}=$$ Can you please give me not the solution of the problem, but ...
3
votes
1answer
60 views

Is there a “smallest” divergent sum? [duplicate]

I'm having a look at analysis right now, and I just thought up this question after reading about the comparison test. Does there exist a "critical" infinite sum of real numbers (which is divergent) ...
1
vote
1answer
39 views

Find the sum of the following series: $\sum_{n=1}^{\infty}\frac{n \cdot 2^n}{(n+2)!}=?$

How can I solve this exercise: Find the sum of : $$\sum_{n=1}^{\infty}\frac{n \cdot 2^n}{(n+2)!}=?$$ I think I should somehow bring the expression to the form of a telescopic one, and make the ...
1
vote
1answer
33 views

Constructing a Continuous Everywhere but Nowhere Differentiable Function

In Neal Carothers' Real Analysis he claims that $$f(x)=\sum_{k \mathop = 0}^\infty 2^{-k}g(2^{k}x)$$ is a continuous but non-differentiable function over the real line if $g(x)$ is the distance ...
2
votes
3answers
54 views

How to find the value of $\sum_{n=2}^{\infty}\frac{3n-5}{n(n^2-1)}$

How can I calculate the sum of this series : $$\sum_{n=2}^{\infty}\frac{3n-5}{n(n^2-1)}=?$$ I've tried to divide in factors $\frac{3n-5}{n(n^2-1)}$ and obtained ...
11
votes
1answer
250 views

Does there exist a closed form for $L_k$ for any $k>3$?

I defined a sequence $L_k$ as the limit of a sequence of "hyperharmonic" series in this question. I was surprised to find that $L_3=(\sqrt{13+4\sqrt2}-1)/2$, but was unable to find a representation ...
1
vote
3answers
54 views

prove the given inequality (for series ) [closed]

For any given $n \in \Bbb N,$ prove that, $$1+{1\over 2^3}+\cdots+{1\over n^3} <{3\over 2}.$$
0
votes
2answers
25 views

To show that $\sum a_nb_n$ is absolutely convergent

Assume that $\sum a_n$ is convergent and $\sum b_n$ is absolutely convergent.To show that $\sum a_nb_n$ is absolutely convergent My try::Consider the sequence of partial sums of $\sum |a_nb_n|$ ...
2
votes
1answer
29 views

A Sequence of Functions Converging to the Derivative at a Point

I'm reading Neal Carothers' Real Analysis and while in the process of constructing an everywhere continuous but nowhere differentiable function, he claims that $$\dfrac{f(v_n)-f(u_n)}{(v_n-u_n)} \to ...
5
votes
5answers
186 views

Prove or disprove: $\sum a_n$ convergent, where $a_n=2\sqrt{n}-\sqrt{n-1}-\sqrt{n+1}$.

Let $a_n=2\sqrt{n}-\sqrt{n-1}-\sqrt{n+1}$. Show that $a_n>0\ \forall\ n\ge1$. Prove or disprove: $\sum\limits_{n=1}^\infty a_n$ is convergent. I can't show that $a_n > 0\ \forall ...
-2
votes
1answer
48 views

Calculate $S =\sum_{k=1}^n\frac {1}{k(k+1)(k+2)}$. [duplicate]

Calculate $S =\sum_{k=1}^n\frac {1}{k(k+1)(k+2)}$. I know I posted this question already but I want a more detailed answer. For example, how you got from one step to another using the partial fraction ...
5
votes
1answer
71 views

Proving $\sqrt{2}=\prod_{n=1}^{\infty }\left(1+\frac{0.75}{4n^2-1}\right)$

Proving $$\sqrt{2}=\prod_{n=1}^{\infty }\left(1+\frac{0.75}{4n^2-1}\right)$$ By using the numerical calculation I saw that the convergence of product series is slow, so I need the proving. thanks.
-1
votes
2answers
29 views

Let $|r| < 1, S = \sum_{k=0}^{\infty} r^k$, and $T = \sum_{k=0}^{\infty} k r^k$. Give a closed form expression for $T$ in terms of $r$. [closed]

Let $|r| < 1, S = \sum_{k=0}^{\infty} r^k$, and $T = \sum_{k=0}^{\infty} k r^k$. Our approach is to write $T$ as a geometric series in terms of $S$ and $r$. Give a closed form expression for $T$ ...
1
vote
0answers
37 views

Partial sum formula for $\sum_{n=0}^{x} {\tan(x)}$ from $(0,\infty)$

I know there is no elementary way of expressing the partial sums of $\tan(x)$. I know; however, I can get an approximation of partial sums using a series, such as the MacLaurin series. If a series can ...
2
votes
1answer
45 views

Sums of Nested Radicals

Let $a_1=\sqrt{2}$, $a_2=\sqrt{2+\sqrt{2}}$ and $a_n$ be defined as $$a_n=\underbrace{\sqrt{2+\sqrt{2+...+\sqrt{2}}}}_{n-times}$$ for any $n\geq1$. Now consider the following infinite sum: ...
2
votes
2answers
57 views

How to prove that $\lim_{n\to\infty}\frac{x_1+2^kx_2+\dots+n^kx_n}{n^{k+1}}=\frac{x}{k+1}$

Knowing that $\lim_{n\to\infty}x_n=x$ I want to prove that $$\lim_{n\to\infty}\frac{x_1+2^kx_2+\dots+n^kx_n}{n^{k+1}}=\frac{x}{k+1}.$$ My guess is that we will use the Stolz–Cesàro theorem. So for ...
2
votes
5answers
123 views

Why does $\sum\limits_{i=1}^n i^2 = An^3+Bn^2+Cn + D$?

I've got this question because of this video (around 3:15). I wonder how setting up a system of $3$ equations will help him solve this problem. I'm thinking I might not understand this because I've ...
3
votes
1answer
19 views

Is this function/series periodic?

$$f(t)=\sum_{k=-\infty}^{\infty}(-1)^kp_{0.5}(t-2k)$$ Recall: $$p_{\Delta}=\begin{cases}\frac{1}{\Delta},&0\leq t\leq\Delta\\0&\text{ otherwise.}\end{cases}$$ Is the function periodic? If ...
3
votes
1answer
60 views

Evaluting sum $\sum_{n=0}^\infty\frac{n^k}{n!}$

Inspired by this question,I was interested if the following sum has a closed form.Looking for $k$ integer I found the Dobinski's formula so that the sum when $k$ is natural number is $e\cdot B_k$ ...
3
votes
4answers
102 views

Calculate $ S =\sum_{k=1}^n\frac {1}{k(k+1)(k+2)}. $

Calculate $S =\displaystyle\sum_{k=1}^n\frac {1}{k(k+1)(k+2)}$. This sequence is neither arithmetic nor geometric. How can you solve this. Thanks!
0
votes
0answers
50 views

Discrete J-method of interpolation (about understanding theorem statement)

The discrete $J$ method is, given Banach spaces $A_0$ and $A_1$: The interpolationn space $[A_0, A_1]_\theta$ is defined by: $a \in [A_0, A_1]_\theta$ if and only if $a$ can be written as ...
1
vote
0answers
75 views

Show that any sequence $(u_{n})$ must tends to infinity as $n→∞$

The motiation to this question can be found in About the solution of a difference equation My question is: Show that any sequence $(u_{n})$ verifying the equation in the above question must tends to ...
1
vote
1answer
48 views

Show that $\lim_{n \to +\infty} \frac{\sum^n x_i}{\sum^n y_i}=a.$ [duplicate]

Let $\{y_n\}$ a sequence so that $\sum y_i=+\infty$ and $$y_n>0, \forall n \in \mathbb{N}.$$ Show that if $$\lim_{n \to +\infty} \frac{x_n}{y_n}=a$$ then $$\lim_{n \to +\infty} ...
1
vote
1answer
72 views

If $a_n$ is prime then $n$ is prime too

Given sequence $(a_n)$ : $a_1=1, a_2=4, a_3=15, a_n=15a_{n-2}-4a_{n-3}$. Prove that if $a_n$ is prime then $n$ is prime too. It is easy to prove that $a_n=4a_{n-1}-a_{n-2}$ and ...
5
votes
1answer
116 views

Limit of this recursive sequence: $x_{n+1}=\bigl(1-\frac{1}{2n}\bigr)x_{n}+\frac{1}{2n}x_{n-1}.$

Consider the following sequence : $x_{0}=a$ , $x_{1}=b$ , $x_{n+1}=\bigl(1-\frac{1}{2n}\bigr)x_{n}+\frac{1}{2n}x_{n-1}.$ Find $\lim_{n\to \infty}x_{n}.$ I calculate $x_{2}$ , $x_{3}$ ,$x_{4}$ ,but ...
1
vote
2answers
36 views

Convergence and limits of recusive sequences.

I want to ask a question about recursive sequences. They have been pretty easy to handle for me, if you have one variable in it. To give you an example, if you have a sequence like: $x_o = 1, ...
-2
votes
2answers
74 views

Sequence of Positive Real Numbers

Let $a_1, a_2, a_3, \ldots$ be a sequence of positive real numbers such that: For all positive integers $m$ and $n$, we have $a_{mn} = a_ma_n$ There exists a positive real number $B$ ...
1
vote
0answers
6 views

How the union of a bound series of integers converges to all integers for cases of all orders.

For the series $S_n = \{-n, \cdots, n \}^d$ I would like to show the union of all such sets converge to $\mathbb{Z}^d$ as $n \rightarrow \infty$. That is to said, how can I prove: $$\bigcup_{n \geq ...
0
votes
1answer
20 views

Alternating Series Convergence Area

I am asked to find for which values my series will converge. $$\sum^{\infty}_{n=0}\frac{x^{5n}}{(4+(-1)^n)^{3n}}$$ after root test I think I can write : when n is even : $$|\frac{x^5}{5^3}| < ...
2
votes
1answer
52 views

Can we simplify this sum?

Let $r>4$ and $n>1$ be positive integers. Can we simplify this sum: $$S=\sum_{m=1}^{n}\frac{2m}{r^{m^2}}$$ I have no idea to start.
0
votes
1answer
22 views

Residue of $g(z)$ at z=0 simple pole

Find the residue of: $$g(z) = \frac{\psi(-z)}{z(z+1)^2} \space \text{at} \space z = 0$$ My Attempt: Because $z=0$ is a simple pole, I thought of using the definition. $$\mathrm{Res} \space _{z=0} ...
1
vote
1answer
25 views

My proof of: Given an adherent point, some sequence converges to it.

(I have two specific questions.) Is my proof correct? Are style, wording, and punctuation alright? $\textbf{Definition 1.}$ Let $X \subseteq \mathbb{R}$. Let $x \in \mathbb{R}$. The point $x$ ...
0
votes
2answers
29 views

Write a sequence that is a geometric and arithmetic progression at the same time.

I thought to write this system $$\begin{cases} y_n = y_1 \cdot q^{n-1} \\ y_n = y_1 + (n+1)d \end{cases}$$ How do I solve it?
0
votes
0answers
17 views

Residue of a rational function

In this answer by Jack D'Aurizio, which is fantastic, I do understand that: If $f(z) = (\psi(-z) + \gamma)^2$ where $\psi(-z)$ is digamma, and $H_n$ (following) is harmonic number: $$\mathrm{Res} ...
2
votes
2answers
66 views

If $\lim\limits_{n\to +\infty}x_n=+\infty$ then $\left(\frac{x_n}{x_{n+1}}\right)_{n\in\mathbb{N}}$ converges

As in the title, in an exercise (Elementary Real Analysis by Thomson and Bruckner p.38), we have to prove that if $\lim\limits_{n\to +\infty}x_n=+\infty$ then ...
0
votes
2answers
56 views

$\lim_{{n}\to{\infty}} (n+1)!(e-1-{1\over2}-…-{1\over n!})= ?$

How can I solve this problem: Find the limit of the following sequence: $$\lim_{{n}\to{\infty}} (n+1)!(e-1-{1\over2}-...-{1\over n!})= ?$$ How to solve this using Cesaro-Stolz ? The numerator and ...
-4
votes
0answers
41 views

Listing significant counterexamples on real sequences [closed]

I give here some basic counterexamples regarding reals sequences. The aim of this post is to inventory other significant counterexamples on real sequences. Please provide the ones that you know and ...
1
vote
1answer
63 views

About the solution of a difference equation

Let $r>4$ be a positive integer. Let us consider this difference equation: $$u_{n+1}=(1+r²ⁿ⁺¹)u_{n}-r²ⁿ⁻¹u_{n-1}+2$$ I want to find a closed form, bu I am not able to find the good idea.
4
votes
4answers
131 views

$\sum _{n=1}^{\infty } \frac{1}{n (n+1) (n+2)}$ Understand the representation

$$\sum _{n=1}^{\infty } \frac{1}{n (n+1) (n+2)}$$=$$\frac{1}{2} \left(-\frac{2}{n+1}+\frac{1}{n+2}+\frac{1}{n}\right)$$ $$s_n=\frac{1}{2} ...
0
votes
1answer
31 views

Can someone check my answers to this problem?

This is from Discrete Mathematics and its Applications Definition of recurrence relation from book From my understanding, compounded annually means that every year(annually) the account will ...
1
vote
2answers
73 views

Delta-Epsilon proof of $\lim_{n\to \infty} r^n = 0 [closed]

Hey I've been having some trouble figuring out this problem. Not sure what to do with the restriction on $r$, or really how to go about writing it formally. Our professor assigned this as a problem ...
5
votes
1answer
120 views

Double sum and zeta function

This is a personal research that came to an end , since the results were not those which were being anticipated. I was unable to come up with a solution therefore I post the topic here: Prove (it ...
2
votes
2answers
202 views

Adding two convergent series

If $\sum_{n=1}^{\infty} a_n$ is finite and $\sum_{n=1}^{\infty} b_n$ is also finite, why is it that you can add the two series term by term and get the sum of the two series? Surely this is ...