1
vote
1answer
30 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
46 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
97 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
202 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. ...
3
votes
1answer
76 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
24 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
39 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 ...
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
137 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
42 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
36 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
80 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
38 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
21 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
44 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
65 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
104 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
49 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
76 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
67 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
149 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
187 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
76 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 $$ ...
12
votes
3answers
568 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
22 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
71 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
39 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
53 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
72 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
81 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 $$e^{iθ}+e^{2iθ}+\cdots+e^{niθ}=\frac{e^{iθ}(1-e^{inθ})}{1-e^{iθ}}$$
7
votes
3answers
253 views

How to evaluate $\sum\limits_{k=0}^{n} \sqrt{\binom{n}{k}} $

Can we find $$ \sum_{k=0}^{n} \sqrt{\binom{n}{k}} \quad$$ This problem asked me my friend about a year ago, but I didn't know how to attack problem. Now, I am interesting in solution. Any suggestion? ...
0
votes
2answers
64 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
35 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
108 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
79 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
86 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
41 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
68 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
33 views
1
vote
1answer
50 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
179 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. ...