0
votes
0answers
16 views

Does this recursion/sequence of iterated infinite sums converge?

Let $n,x\in\mathbb{N}$, $\alpha,\beta,\lambda\in\mathbb{R}^+$, where $\alpha,\beta<1$. Does the following sequence converge (and to what)? $s_0=\alpha n+\lambda$ ...
1
vote
0answers
26 views

Extracting coefficients from a transformed generating function

Let $G(z)=\sum_{k\geq 0} a_kz^k$ be a generating function such that $z^aG(1-z)=P(z)$, where $P(z)$ is a polynomial and $a$ is a positive integer. I'm interested in $P(z)[z^n]$, the coefficient in ...
1
vote
1answer
35 views

Problem Relating to Error in Series

For the following series, find the number of terms required to find the sum with error < 0.005, and find upper and lower bounds for the sum using a much smaller number of terms. ...
2
votes
2answers
74 views

How to prove that $\sum_{n}^{\infty}u_{n}^{2}\left(u_{1}+\cdots+u_{n}\right)^{-1}<\infty$ under these conditions?

Suppose that $u_n$ is a decreasing sequence of positive numbers that converges to zero. Suppose moreover that $S_n = \sum_{k=1}^n u_k$ diverges. I would like to prove that the sum $\sum u_k^2 / S_k$ ...
0
votes
1answer
31 views

Convergence of the $\sum_{1}^\infty n^{\frac1n}$

I stumbled upon that question tried some tests like ratio test and ahare test but all gave the limit 1 which is indecisive. Any one with a better approach..
1
vote
1answer
19 views

Convergence of two unusual “nested” sums

I was contemplating convergent sums, trying to think of very unusual or unorthodox sums that might be treatable recursively. Eventually, the following sum occurred to me: $$ \xi = 1 + \frac{ ...
0
votes
1answer
41 views

Prove there is a subsequence $(a_{nk})_{n=1}^\infty$ such that $\Sigma^{\infty}_{k=1} a_{nk}$ converges.

Hey everyone this was give as a practice problem and i'm having trouble, any help is appreciated Let $(a_n)_{n=1}^\infty$ be a sequence such that $\displaystyle \lim_{n \rightarrow \infty} {a_n} = ...
4
votes
2answers
62 views

Is $f(x)=\sum_{k\in\mathbb N}\frac1k\sin\frac x{2^k}$ bounded?

$$f(x)=\sum_{k\in\mathbb N}\frac1k\sin\frac x{2^k}$$Is this function bounded? So obviously this converges because $|\frac1k\sin\frac x{2^k}|<|\frac x{2^k}|$ and $\sum\frac x{2^k}$ converges by ...
0
votes
1answer
28 views

Is $f$ integrable in $L(X,\mathcal{X},\mu)$

Is $f$ integrable $L(X,\mathcal{X},\mu)$ $\mu(E)=\sum_{n\in E\cap\mathbb{N}} |n^2+n-6|$ $f:\mathbb{R}\rightarrow \mathbb{R_+}\cup\infty$ $f=(x-2)^{-4}$
3
votes
1answer
87 views

Proof that $0 < \lim(a_n/b_n) < \infty$ implies convergence/divergence of $a_n$ and $b_n$

Suppose that $a_n, b_n > 0$ for all $n$. Prove that if $0 < \lim(a_n/b_n) < \infty$, then $\sum a_n$ and $\sum b_n$ either both converge or both diverge. I'm a bit unsure on how to proceed ...
1
vote
1answer
42 views

$\sum$ over disjoint union of sets

In Discrete mathematics, rule of sum says that "If a first task can be performed in $m$ ways and another can be performed in $n$ ways and two task be independent, then whole work can accomplished in ...
0
votes
3answers
59 views

Using Cauchy-Schwarz inequality to prove that the mean of n real numbers is less than or equal to the root-mean-square of those numbers

Expressed mathematically, the question is to prove the that $\frac{1}{n}$ $\sum_{i=1}^{i=n}{a_i}\leqslant$ $\sqrt{\frac{1}{n}\sum_{i=1}^n{x_i}^2}.$ First of all, what form of Cauchy-Schwarz should I ...
3
votes
2answers
56 views

Showing that $\sum \sqrt{a_na_{n + 1}}$ converges given that $\sum a_n$ converges [closed]

Suppose a series $\sum a_n$ of nonnegative reals converges; show that $\sum \sqrt{a_na_{n + 1}}$ also converges.
2
votes
0answers
38 views

calculating sum of a limit of integral

I am trying to calculate the following expression $$ \sum_{m=0}^{\infty} \frac{1}{m!} \lim_{n \to \infty} \int_{\{(x,y):2x^2+y^2<n^2 \}}\left( 1 - \frac{2x^2+y^2}{n^2}\right)^{n^2} x^{2m}dx~dy ...
11
votes
3answers
132 views

How prove this limit $\lim_{n\to\infty}\frac{1}{n}\sum_{i=1}^{n}\sum_{j=1}^{n}\frac{i+j}{i^2+j^2}=\frac{\pi}{2}+\ln{2}$

show that: this limit $$I=\lim_{n\to\infty}\dfrac{1}{n}\sum_{i=1}^{n}\sum_{j=1}^{n}\dfrac{i+j}{i^2+j^2}=\dfrac{\pi}{2}+\ln{2}$$ My try: ...
3
votes
4answers
123 views

Determine if $\sum_{n=1}^{\infty}\frac{(-1)^nn^2+n}{n^3+1}$ converges or diverges.

Another series I found I'm struggling with. Determine if the following series converges or diverges.$$\sum_{n=1}^{\infty}\frac{(-1)^nn^2+n}{n^3+1}$$ Ratio test and n-th root test are both ...
2
votes
1answer
69 views

The limit of a sum with Taylor's theorem

I am trying the fallowing exercise : Let $f\in C^1(\mathbb{R},\mathbb{R})$ with $f(0)=0$ Compute the limite of : $S_n=\sum _{k=0}^{n}f(\frac{k}{n^2})$ I used Taylor's theorem , so I have $$ ...
10
votes
3answers
445 views

an inequality: $1+\frac1{2^2}+\frac1{3^2}+\dotsb+\frac1{n^2}\lt\frac53$

$n$ is a positive integer, then $$1+\frac1{2^2}+\frac1{3^2}+\dotsb+\frac1{n^2}\lt\frac53.$$ please don't refer to the famous $1+\frac1{2^2}+\frac1{3^2}+\dotsb=\frac{\pi^2}6$. I want to find a ...
1
vote
0answers
19 views

nonlinear sequence to sequence transformations

i know matrix methods such as Cesaro,Holder,Riesz are regular linear sequence transformations. i wonder if there is any regular nonlinear sequence transformation?
2
votes
2answers
69 views

Help with compact notation for sum

I've already understood the motive of this sum using the nom-compact way, but I want to do it in the compact way, so it will be rigorous. Please, I need some help: ...
1
vote
1answer
35 views

is this or (when) does this equality hold for weighted power series

$s_n=\sum _{k=0}^n a_k$ for every $n$ and $x\in(0,1)$. Let $p=(p_n)$ is a sequence of nonnegative numbers with $p_0>0$ s.t $P_n= \sum_{k=0}^{n}p_k \rightarrow \infty$ as $n \rightarrow \infty$ ...
1
vote
2answers
52 views

Determine whether $\sum_{n=1}^{\infty} \frac{x^2}{n^2}$ converges uniformly on $[5,\infty)$

I cannot figure out whether or not $\sum_{n=1}^{\infty} \frac{x^2}{n^2}$ converges uniformly over $[5,\infty)$. My first thought was to try using the Weierstrass M-Test but failed immediately. Is ...
0
votes
3answers
70 views

Is a sequence convergent and if so what is the sum

The sum $$\sum_{n=1}^{\infty} \frac{1}{(n+3)(n+2)} $$ Ive made it into partial fractions which gives $\frac{1}{n+2} - \frac{1}{n+3}$ But im unsure how to tell if this now converges as obviously as ...
2
votes
3answers
74 views

Prove that $2\sum_{k=1}^n \cos(kθ) = \frac{\sin[\left(n+1/2\right)θ]}{\sin(θ/2)}-1$ [closed]

Prove that $$2\sum_{k=1}^n \cos(kθ) = \frac{\sin[\left(n+1/2\right)θ]}{\sin(θ/2)}-1$$ By using $$e^{iθ}+e^{2iθ}+\cdots+e^{niθ}=\frac{e^{iθ}(1-e^{inθ})}{1-e^{iθ}}$$
0
votes
2answers
63 views

Find an index n such that inequality is true

I need to find an index $n$ such that: $|e^{x} - S_{n}(x)| \leq \frac{|e^{x}|}{10^{4}}$ (1), where $S_{n} = \sum\limits_{i=0}^{n} \frac{x^{k}}{k!}$ is the n-th Partial Sum of $e^x$. Let $x$ be a ...
2
votes
0answers
34 views

Show that $ \lim_{n\to\infty}\sum_{k=1}^{n}\frac{n}{n^2+k^2}= \frac{\pi}{4}$ [duplicate]

Show that: $$ \lim_{n\to\infty}\sum_{k=1}^{n}\frac{n}{n^2+k^2}= \frac{\pi}{4}$$ My idea: I thought that this could be rewritten as an integral, then use trig substitution (perhaps tangent) and then ...
1
vote
1answer
75 views

Summation of trigonometric functions

So consider a summation of ai cos (x + phi_i) where i ranges from 1 to N. Could we describe this summation as a single cosine function? Or the sum of two cosine or sine functions? How would we do this ...
1
vote
2answers
74 views

Determine the value of $p>0$ for which $\sum_{n=1}^{\infty}(-1)^{\lfloor{\sqrt{n}}\rfloor}/n^p$ converges.

Determine the value of $p>0$ for which $$\sum_{n=1}^{\infty}\frac{(-1)^{\lfloor{\sqrt{n}}\rfloor}}{n^p}$$ converges. By considering $\lfloor{\sqrt{n}}\rfloor$, we see the series is $$\sum_{k\ge1} ...
0
votes
3answers
53 views

Prove $\sum_{i = 0}^{n} { (3 i+1)^2}$ is $(n+1) (6 n^2+9 n+2)/2$?

I am stuck on what kind of process to use to go about solving this problem: Prove that $\sum_{i = 0}^{n} { (3 i+1)^2}=(n+1) (6 n^2+9 n+2)/2$ Any advice would be great! Thanks so much!
1
vote
1answer
56 views

Where to find a proof of Silverman-Toeplitz?

I am referring to the theorem which gives a necessary and sufficient condition on a infinite matrix that maps convergent sequences to sequences converging to the same limits. Wiki gives a link to ...
1
vote
1answer
36 views

Getting close to any real with a semi-convergent series

I got a really intuitive exercise here, but I'm having a hard time actually fulfilling a proof. Let $\sum_{0}^{}a_n$ be a semi-convergent real series. Let $l \in \mathbb R$. Show that there exists ...
1
vote
0answers
64 views

Proof of absolutely convergent sums over two indices.

In the book Concrete Mathematics (2nd) written by Ronald Graham, Donald Knuth and Oren Patashnik, they prove the next theorem. Absolutely convergent sums over two or more indices can always be ...
2
votes
0answers
30 views
1
vote
1answer
49 views

Question about sums and double sums

I want to prove the following theorem: Suppose that we have given $s_{ij}\in[0,\infty]$ for each $i,j\in\mathbb{N}$. Define the sequence $(t_n)$ by: $t_1=s_{11}, ...
6
votes
1answer
166 views

proof that $f(x)=\sum_{n=1}^\infty \frac{\ln(x+n)}{x^2 + n^2}$ converges uniformily

Prove that $f(x)=\sum_{n=1}^\infty \frac{\ln(x+n)}{x^2 + n^2}$ converges uniformly for $x\geq 0$. This exercise is in a text about uniform convergence, I've tried to use Weierstrass test with no ...
2
votes
2answers
74 views

$\sum_{n=1}^\infty (1 - \cos \frac1n)^\alpha\log n$

How do I find for which values of $\alpha \in \mathbb{R}$ the sum converges? $$\sum_{n=1}^\infty (1 - \cos \frac1n)^\alpha\log n$$ I have tried using the following techniques: Comparison test. ...
0
votes
1answer
72 views

Why is this summation equals $1$?

Referring to the conditions in parenthesis, why is the summation expression in the last line equal to $1$? (We may also assume that $-1< s < 1$.)
2
votes
1answer
218 views

Beppo-Levi's Theorem

Considering $\sum_{j = 0}^{\infty} \int f_j(x) < \infty$ and $\sum_{j = 0}^{\infty} \int g_j(x) < \infty$, how can I simplify the following expression ? $$ \int \frac{\sum_{j = n}^N ...
4
votes
1answer
86 views

What's $\sum_{n \ge 0} q^{n^2}$?

Is there a relatively simple way of calculating the sum of the series $$\sum_{n=0}^\infty q^{n^2}, \quad |q|<1 ?$$
0
votes
2answers
37 views

Finding the sequence that maximizes a constrained sum

Let $0 < a < 1$ and let $S_k$ be a unknown sequence of such that $S_k > 0$ and $$ S_n + S_{n-1} + \ldots + S_1 = C = constant. $$ What should be $S_k$ so that the sum $$ S_n + aS_{n-1} + ...
2
votes
2answers
1k views

What is sum of reciprocal? [duplicate]

Is ${1\over 1}+{1\over 2}+{1\over 3}+\cdots +{1\over n}$ computable? I couldn't find any formulas to find the result. The explanation would be very helpful. Thanks before.
4
votes
2answers
377 views

Sum : $\sum \sin \left( \frac{(2\lfloor \sqrt{kn} \rfloor +1)\pi}{2n} \right)$.

Calculate that : $$ \sum_{k=1}^{n-1} \sin \left( \frac{(2\lfloor \sqrt{kn} \rfloor +1)\pi}{2n} \right).$$
3
votes
1answer
172 views

Asymptotics of a summation over real valued functions

Let $f$ and $g$ be integrable in $[0,1]$ and $(-\infty, \infty)$ respectively. Let $a_k$ be a divergent series of positive terms and $S_k = a_1 + a_2 + \ldots + a_k$ such that the following ...
1
vote
1answer
170 views

Can anyone see why this lemma is true? Seems very confusing!

Given a function f, we write $\bar{f}(t) = \sup_{u \leq t} f(u).$ Lemma: Let $s_{0},\dotsc,s_{T}$ be real numbers and $h:\mathbb R \to \mathbb R$. Then ...
0
votes
5answers
68 views

What is $\sum_{j=0}^{\infty}\frac{j+1}{2^j}$?

What is $\sum_{j=0}^{\infty}\frac{j+1}{2^j}$? I don't know what theorems or tricks to use to add up an infinite sum.
2
votes
1answer
44 views

Showing that a sum diverges

Suppose that $a_{j} \geq 0$ and that $\sum a_{j}$ diverges. Prove that $\sum\frac{a_{j}}{1+a{j}}$ diverges. The hint that is given is show that it if it converges $a_{j} \rightarrow 0$. I don't ...
0
votes
0answers
58 views

Strange convolution equation

In an article ( https://www.dropbox.com/s/3012v4s1ngpimvg/gridding_Schomberg_Trimmer.pdf ) about implementation of Gridding method for parallel-beam tomography there's an equation(#47 in the article): ...
4
votes
1answer
67 views

Limit of difference of integral and sum

$f:[0,1]\rightarrow\mathbb R$ and $f\in C^1$, then the limit $\lim_{n\rightarrow\infty} n(\int_{0}^{1}f(x)dx-\frac{1}{n}\sum_{k=1}^{n}f(\frac{k-1}{n}))$ exists. I guess the kernel lies in the sum ...
6
votes
3answers
1k views

Summation Symbol: Changing the Order

I have some questions regarding the order of the summation signs (I have tried things out and also read the wikipedia page, nevertheless some questions remained unanswered): Original 1. wikipedia ...
1
vote
0answers
73 views

Pi identity with sum and product

Please prove this identity $$\sum_{ n=1 }^{\infty }\left({\left(-1\right) }^{ n }\frac{\prod_{ j=1 }^{ n }{\left(\frac{ 3 }{ 2 }-j\right) }}{\left( 2n+1\right)\left( n!\right) }\right) =\frac{\pi }{ ...