1
vote
1answer
30 views

Relationship between $\int_a^b f(x) dx$ and $\sum_{i= \lceil a\rceil}^{\lfloor b\rfloor} f(i)$

Let we have a continuous function $f(x)$ in the interval $ [ a,b ] $ Does there exist any relationship between its integral and summation of function-values defined at the integers between $a$ and ...
0
votes
0answers
29 views

Inequality in $\mathbb{Z}^2$

Let $k=(k_1,k_2)\in\mathbb{Z}^2$. Denote $|k|\leqslant n$ when $|k_1|,|k_2|\leqslant n$. I need help to show $$|\sum_{k+l+m=0}_{|k|,|l|,|m|\leqslant ...
0
votes
2answers
53 views

Using induction to prove a formula for $\sin x+\sin 3x+\dots+\sin (2n-1)x$

I'm working from the text "Intro To Real Analysis" by William Trench. Here is what I have thus far. I will prove using Mathematical Induction that $\sin x+\sin 3x+...+\sin (2n-1)x=\frac{1-\cos ...
1
vote
1answer
43 views

Value of $\psi\left(\frac{1}{2}\right)$

I apologise if this is a dumb question, but I have trouble deriving $\displaystyle\psi\left(\frac{1}{2}\right)=-\gamma-2\ln{2}$. I have tried the following. \begin{align} \psi\left(\frac{1}{2}\right) ...
3
votes
2answers
125 views

Is there any closed form for the finite sum $1+\dfrac{1}{2}+\dfrac{1}{3}+\dfrac{1}{4}+…+\dfrac{1}{n}?$ [duplicate]

I know that the infinite summation $$1+\dfrac{1}{2}+\dfrac{1}{3}+\dfrac{1}{4}+...+\dfrac{1}{n}+...$$ is divergent and also the sequence $$1+\dfrac{1}{2}+\dfrac{1}{3}+\dfrac{1}{4}+...+\dfrac{1}{n}-\ln ...
7
votes
1answer
123 views

Showing that $\sum_{i=1}^n \frac{1}{|x-p_i|} \leq 8n \left( 1 + \frac{1}{3} + \frac{1}{5} + \cdots + \frac{1}{2n-1} \right)$

I'm taking a summer analysis course and preparing for our final exam later this week. Our professor gave us the following problem on our mock exam, and I can't seem to get anywhere on it. Does anyone ...
-4
votes
2answers
100 views

What does this infinite sum converge to?: $\sum_{n=1}^\infty \dfrac{1}{n^k} = \dfrac{1}{1^k} + \dfrac{1}{2^k} + \dfrac{1}{3^k} + …$

$$\sum_{n=1}^\infty \dfrac{1}{n^k} = \dfrac{1}{1^k} + \dfrac{1}{2^k} + \dfrac{1}{3^k} + \dfrac{1}{4^k} + \dfrac{1}{5^k} + ...$$ I've found that: when $k=1$, it diverge to infinity when $k=2$, it ...
0
votes
0answers
50 views

Sigma Notation Inequality

Given two sets of nonnegative real numbers: $$\{a_1, a_2, ..., a_N\}, \{b_1, b_2, ..., b_N\}$$ Are there any conditions for which the following inequality is true? $${1\over N} \sum_{i=1}^N ...
1
vote
1answer
29 views

Transforming a power tower to a product

It is possible to write the product of a sequence of terms $a_i$ as a function of the sum of a sequence of functions of these terms: $$\prod_i a_i=f\left(\sum_i g(a_i)\right)$$ where $f=\exp$ and ...
2
votes
0answers
28 views

A question about the differentiability of two Weyl sums

Consider the following functions, associated with certain trigonometrical sums: $$ f_{\alpha,\beta}(x) = \sum_{n=1}^{+\infty}\frac{\cos(n^{\alpha+\beta}x)}{n^{\alpha}},\qquad g_{\alpha,\beta}(x) = ...
3
votes
1answer
101 views

find a $B_{n,j}$ such that $|A_{n,j}-L_j| \leq B_{n,j}$ $\forall n,j$ and $\sum_{j=0}^{\infty}B_{n,j}$ converges

We have $A_{n,j}= 3(-1)^j2^{n-j+1}\frac{(2(n-j)-4)!}{(n-j)!(n-j-2)!}\binom{j+2}{2}\frac{n^\frac{5}{2}}{8^n}$ and $L_j=(-\frac{1}{8})^j\binom{j+2}{2}\frac{3}{8\sqrt{\pi}}$ So I know $\lim_{n \to ...
1
vote
1answer
31 views

evaluating a sum using Cauchy condensation test

Let $$\sum\limits_{n\ge1}{\frac{(-1)^n}{n^\alpha \ln n}}$$ I want to check if the sum is converges absolutely. Hence, we need to check the convergence of $$\sum\limits_{n\ge1}{\frac{1}{n^\alpha \ln ...
3
votes
1answer
48 views

limit of a sum of powers of integers [duplicate]

I ran across the following problem in my Advanced Calculus class: For a fixed positive number $\beta$, find $$\lim_{n\to \infty} \left[\frac {1^\beta + 2^\beta + \cdots + n^\beta} {n^{\beta + ...
0
votes
1answer
102 views

How does one graph $\sum_{x=0}^{n}$ [closed]

How does one graph a summation, like $$\sum_{x=0}^{n} n$$ Can it be like this Because if you take the points from the summation (0,0), (1,1), (2,3), (3,6) you can tell by summations it only works ...
1
vote
1answer
37 views

defenite integral involve bessel function

I have an integral which involves Bessel function as follows: $I=\int_{r=0}^a \int_{\theta=0}^{2\pi}(e^{-jkr\cos(\theta-\phi)}d\theta)rdr$ I have tried with $e^{-jkr\cos(\theta-\phi)}=\sum ...
7
votes
1answer
208 views

Series $\sum \frac{1}{n^2\sin^3n}$

Question : Show that series $\sum \cfrac{1}{n^{2}\sin^{3}n}$ is divergent. Hint: Show that $$\sum \frac{1}{n|\sin(n)|}$$ is divergent. I am interested in other possible proofs for this question. ...
6
votes
2answers
100 views

Sum Involving Bernoulli Numbers : $\sum_{r=1}^n \binom{2n}{2r-1}\frac{B_{2r}}{r}=\frac{2n-1}{2n+1}$

How can we prove that $$\sum_{r=1}^n \binom{2n}{2r-1}\frac{B_{2r}}{r}=\frac{2n-1}{2n+1}$$ where $B_{2r}$ are the Bernoulli numbers? $$\begin{array}{c|c|c|} n & \frac{2n-1}{2n+1} & ...
3
votes
1answer
82 views

Asymptotic of a sum evaluation as $ x \to \infty $

Let be the sum $$ \sum_{n\le x}[x/n]=g(x) $$ where $ [x] $ means floor function. My best try for asymptotic is $ g(x) \sim x\log (x)+\gamma x +1$ where I have used the asymptotic $ [x] \sim x $ ...
2
votes
0answers
27 views

Interchanging index of summation in $d$ dimensions

Let $\alpha = (\alpha_{1}, \ldots, \alpha_{d}) \in \mathbb{Z}_{\geq 0}^{d}$ and let $|\alpha| = \alpha_{1} + \cdots + \alpha_{d}$. I have the following question about interchanging summations: Is ...
0
votes
1answer
46 views

What does it mean for a series to be convergent?

I have the definition: Let $(a_n)$ be a sequence of real numbers. Let $s_n=a_1+a_2+...+a_n$. We say the series $a_1+a_2+...$ is convergent if the sequence of partial sums $(s_n)$ is convergent. The ...
1
vote
1answer
92 views

Show that if $\sup\big\{\sum\lvert\, f(a)\rvert\big\} < \infty$, then $\{ a \in A : f(a) > 0\}$ is countable.

Let $f:A \to \mathbb R$ and suppose that $$ \sup\Big\{\sum_{a\in F}\lvert\, f(a)\rvert : F\text{ is finite subset of }A\Big\} < \infty $$ then the set $\{ a \in A : f(a) > 0\}$ is countable. ...
0
votes
1answer
21 views

Using M-Test to show you can differentiate term by term.

I have the series $\sum_{n=1}^\infty \frac{\lambda^{n-1}n}{n!}=\sum_{n=1}^\infty \frac{d}{d\lambda}\big(\frac{\lambda^n}{n!} \big)$ and I would like it to be $\frac{d}{d\lambda}\big(\sum_{n=1}^\infty ...
4
votes
0answers
171 views

Integral $I=\int_0^1 \frac{\log x \log (1+x) \log(1-x) \log(1+x^2)\log(1-x^2)}{x^{3/2}}dx$

Hi I am trying to integrate and obtain a closed form result for $$ I:=\int_0^1 \frac{\log x \log (1+x) \log(1-x) \log(1+x^2)\log(1-x^2)}{x^{3/2}}dx. $$ Here is what I tried (but I do not think this is ...
2
votes
3answers
43 views

Expression generating $\left( \frac{3}{10}, \, \frac{3}{10} + \frac{33}{100}, \, \frac{3}{10} + \frac{33}{100} + \frac{333}{1000}, \dots \right)$

I'm looking for a closed-form expression (in terms of $n$), that will give the sequence $$ (s_n) = \left( \frac{3}{10}, \, \frac{3}{10} + \frac{33}{100}, \, \frac{3}{10} + \frac{33}{100} + ...
2
votes
0answers
43 views

Lower bound of $\sum_{k = 1}^{N}1/(x + k)$

Let $f(x) := \sum_{k = 1}^{N}1/|x + k|$ for $x \in [0, N]$. Why is $f(x) \geq C\log N$ for all $x \in [0, N]$ where $C$ is an absolute constant. My work is: Since $x \in [0, N]$, we can remove the ...
2
votes
1answer
82 views

$\sum_{n=-\infty}^\infty e^{-\alpha n^2+\beta n}$

Hi I am trying to calculate the sum given by $$ \sum_{n=-\infty}^\infty e^{-\alpha n^2+\beta n}=\ = \sqrt{\frac{\pi}{\alpha}} e^{\beta^2/(4\alpha)} ...
0
votes
0answers
24 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
39 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
36 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
103 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
32 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
27 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
45 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
66 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
31 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
106 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
52 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
105 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
68 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
42 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
158 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
218 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
77 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 $$ ...
14
votes
3answers
640 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
24 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
73 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
41 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
54 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
73 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 ...
3
votes
3answers
95 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 $$e^{iθ}+e^{2iθ}+\cdots+e^{niθ}=\frac{e^{iθ}(1-e^{inθ})}{1-e^{iθ}}$$