Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [sequences-and-series]

For questions concerning sequences and series. Typical questions concern, but are not limited to: recurrence relations, convergence tests, identifying sequences, identifying terms. For questions on finite sums, use the (summation) tag instead.

0
votes
1answer
30 views

Local isolation of zeros complex analysis

I've been studying from Basic complex analysis by Mardsen and Hofmann, and I came across the proposition and corollary on page 212: Proposition 3.2.9 Suppose $f: \Omega \rightarrow \mathbb{C}$ is ...
-6
votes
2answers
46 views

prove $\sum_1^\infty (-1)^k / \sqrt{k}$ converges [closed]

How could one prove that $$\sum_1^\infty (-1)^k / \sqrt{k} $$ converges?
27
votes
5answers
814 views

Evaluating $\int^1_0 \frac{\log(1+x)\log(1-x) \log(x)}{x}\, \mathrm dx$

In this thread a friend posted the following integral $$I=\int^1_0 \frac{\log(1+x)\log(1-x) \log(x)}{x}\, \mathrm dx$$ The best we could do is expressing it in terms of Euler sums $$I=-\frac{\...
1
vote
1answer
56 views

Is there a way to evaluate this integral without power series?

I want to evaluate $$ \int \frac{\arctan(x)}{x} dx $$ One way of doing it is to use power-series. I can re-write $$ \arctan(x) = \sum_{n=0}^\infty (-1)^n \frac{x^{2n+1} }{2n+1} \ \ |x|<1 $$ $$ \...
0
votes
1answer
19 views

Find first two terms of sequence where the relationship of the first three terms is known

The sequence $u_n$ is defined by: $u_{n+2}=2*u_{n+1}+2*u_n$ for $n\geq2$ with given $u_0$ and $u_1$ $u_0$ = 3 $u_1$ = 3 My goal is to find $u_2$ and $u_3$ but I'm struggling to see how I can go ...
0
votes
2answers
41 views

General formula of partial sum of series (non-geometric)

Like user71317 in his question I am struggling to understand how we arrive at the general formula of the partial sums of a series. In my case the following series: $$\sum_{n=2}^{\infty} \frac{1}{n^2-...
2
votes
3answers
228 views

Radius of convergence of $\sum_{k = 0}^{\infty}\frac{ k^{2 k + 5} \ln^{10} k \ln \ln k}{\left(k!\right)^2} x^k$

I want to find the radius of convergence of $$\sum_{k = 0}^{\infty}\frac{ k^{2 k + 5} \ln^{10} k \ln \ln k}{\left(k!\right)^2} \,x^k$$ I know formulae $$R=\dfrac{1}{\displaystyle\limsup_{k\to\...
0
votes
0answers
17 views

Dirichlet series on positive reals [closed]

I consider a séquence $(a_{n})$ of positive real numbers, and $f:s \rightarrow \sum_{n \geq 1} \frac{a_{n}}{n^{s}}$. The serie converges absolutely for $\Re(s) \geqslant 0$ and the function f has a ...
0
votes
1answer
17 views

Algorithm for evaluating a sequence while removing from that sequence?

Introduction : I'm looking at this problem and I've gotten stuck on it. The rules are as follows: Problem Definition : You have a sequence of numbers -- 1 through x (where x > 1). Two numbers are ...
2
votes
4answers
69 views

Evaluating $\sum_{k=1}^{n} \frac{1}{k(k+1)}$

I have just started learning sums. I need to evaluate the following sum: $$S_{n} = \sum_{k=1}^{n} \frac{1}{k(k+1)}$$ $$a_{1} = \frac{1}{2}, a_{2} = \frac{1}{6}, a_{3} = \frac{1}{12}, a_{4} = \frac{1}...
-1
votes
5answers
71 views

Find $\lim\limits_{n\to\infty}x_n$ [closed]

Where $$x_{n+1}= \begin{cases} \frac{x_n}{2} & \text{if } x_n \text{ is even} \\ \frac{1+x_n}2 & \text{if } x_n \text{ is odd} \end{cases}$$ and $x_0\gt0$ and $n\geq0$. Any idea on how to ...
0
votes
2answers
71 views

Proving integral $\int_0^1\frac{e^x-1}{x}$ is equal to $\sum_{n=1}^{\infty}\frac{1}{n \cdot n!}$

Show that the following equality is true. $$ \int_0^1\frac{e^x-1}{x}\, \mathrm dx = \sum_{n=1}^{\infty}\frac{1}{n \cdot n!} $$ How can I tackle this problem?
2
votes
2answers
2k views

Is the sequence of function $\frac{\sin nx}{\pi x}$ convergent to $\delta(x)$

I have no doubt that the integral of the function from negative infinity to positive infinity is 1. But I don't think $f_n(x) = \frac{\sin nx}{\pi x} \rightarrow 0 $ as $ n \rightarrow \infty $ given $...
0
votes
0answers
20 views

The remainder term of a maclaurin series

Suppose $f$ has a local minimum at $0$, i.e. $f'(0) = 0, f''(0)>0$, and suppose $f''$ is continuous in some interval $[0,T]$ ,$T>0$, with $f'' > 0 $ in this interval. Then we must have that $...
0
votes
1answer
40 views

numbers which are not obtained by the given equation

I have an equation : $y = 10a+11b+12c$ Here atleast two of coffecients $a,b,c$ are >=1, the others can be 0. For example: $ a= 1, b= 1, c= 0 $ so $y$ becomes 21. What are set of natural numbers ...
0
votes
1answer
32 views

Show that there is $x\in[-1;1]$ such as the set of limit point of $u_n(x)$ is $[-1;1]$ where $u_0(x) = x, u_{n+1}(x) = 2u_n(x)^2 - 1$

Let be $f$ defined by $f(x) = 2x^2 -1$. Let be $(u_n(x))$ the sequence defined by: $u_0(x) = x, u_{n+1}(x) = f(u_n(x))$ Show that there is $x\in[-1;1]$ such as the set of limit point of $u_n(x)$ is ...
1
vote
2answers
28 views

The limit $\lim_{n\to \infty}f(2^{-n}x)$ for different $x$.

Suppose $f:\Bbb R\backslash \{0\} \to\Bbb R$ is continuous and that for each $x\in\Bbb R\backslash \{0\}$ the limit $$\lim_{n\to \infty}f(2^{-n}x)$$ exists. Is it true that the limit must have the ...
2
votes
1answer
41 views

Calculate $\sum_{n=-\infty}^{\infty}\frac{1-\cos(an)}{(an)^2}$

After playing with some series in a numerical math website, it seems to me like the following identity holds: $$\sum_{n=-\infty}^{\infty}\frac{1-\cos(an)}{(an)^2}=\frac{\pi}{a}$$ It seems a little ...
1
vote
1answer
46 views

nth term of the sequence 1,2,3,5,7,9…

what will be the formula for finding the nth term of a series in which the difference between the terms increase by 1 after every k elements For example (for k = 3) : ...
2
votes
1answer
47 views

Prove that $\sum_{n=1}^{\infty}\log \cos \left (\frac{1}{n}\right )$ converges absolutely.

Prove that $$\sum_{n=1}^{\infty}\log \cos \left (\frac{1}{n}\right )$$ converges absolutely. The answer here suggests to use the Limit Comparison Test but it works for $a_n \geq 0$ while $\ln(\cos (1/...
11
votes
2answers
165 views

Find $\sum_{n=1}^{\infty}\int_0^{\frac{1}{\sqrt{n}}}\frac{2x^2}{1+x^4}dx$

I want to find the sum: $$\sum_{n=1}^{\infty}\int_0^{\frac{1}{\sqrt{n}}}\frac{2x^2}{1+x^4}dx$$ I start with finding the antiderivative of the integrant, which is: $$\frac{1}{2\sqrt{2}}[\ln(x^2-\...
0
votes
1answer
43 views

$\sum_{n=1}^\infty \frac{a_1+a_2+\cdots+a_n}{n}$ converges/divereges?

Let $a_n\geq 0$. $\sum_{n=1}^\infty \frac{a_1+a_2+\cdots+a_n}{n}$ converges/divereges? Take $a_n=1/n$ for example, then the proposed series looks like $\sum\frac{\ln n}{n}$, which is divergent. What ...
1
vote
3answers
39 views

Prove the divergence of $\sum_{k=1}^{\infty}{\frac{k}{\sqrt[3]{k+1}}}$ (Alternative proofs)

I argued like that:$$ \lim\limits_{k\to\infty}\frac{k}{\sqrt[3]{k+1}}=\lim\limits_{k\to\infty}\frac{\sqrt[3]{k}\cdot \sqrt[3]{k^2}}{\sqrt[3]{k}\cdot \sqrt[3]{1+\frac{1}{k}}}=\lim\limits_{k\to\infty}\...
1
vote
1answer
27 views

$\sum_{n=1}^\infty (a^\frac{1}{n}-\frac{b^\frac{1}{n}+c^\frac{1}{n}}{2})$ convergent or divergent?

Is $$ \sum_{n=1}^\infty \left(a^\frac{1}{n}-\frac{b^\frac{1}{n}+c^\frac{1}{n}}{2}\right) $$ convergent or divergent? Here, $a,b,c>0$. It is clear that the term tends to $0$. But how do I ...
1
vote
1answer
34 views

What is $\lim_{n \to \infty} \big[ \frac{1}{a_{1}a_{2}}+\frac{1}{a_{2}a_{3}}+…+\frac{1}{a_{n-1}a_{n}}\big]$?

Let $a_{1}=1$, $a_{n}=a_{n-1}+4, \ \forall n \geq 2$ then what is $$\lim_{n \to \infty} \big[ \frac{1}{a_{1}a_{2}}+\frac{1}{a_{2}a_{3}}+...+\frac{1}{a_{n-1}a_{n}}\big]$$ Using recurrence relations I ...
0
votes
1answer
75 views

Are there infinitely many prime numbers in the Look-and-say sequence?

Question: Are there infinitely many prime numbers in the Look-and-say sequence? In the sequence, I found that $11$, $312211$ and $13112221$ are prime numbers.
-2
votes
0answers
33 views

is there any equation to find number of days in a particular month [closed]

Can anyone generalise the number of days in that particular month in the form of an equation
3
votes
1answer
196 views

Is there a closed form for a give infinite sum?

I've been asked to evaluate this sum $$\sum_{n=0}^{\infty}\frac{C_n^2}{2^{4n}}(H_{n+1}-H_n)$$ where $C_n=\frac1{n+1}{2n\choose n}$ denotes the $n$th Catalan number and $H_n$ denotes the nth Harmonic ...
3
votes
1answer
33 views

Limit of ratio of consecutive terms of a recurrence relation.

My question is: The sequence $a_n$ is defined by $$\sum_{r=1}^{k}p_ra_{n+r}=0,\forall n$$ where $p_r$ are fixed constants, and the "initial conditions" (i.e. the given values of $a_1,\ldots,a_{k}$)....
1
vote
2answers
46 views

Which one of the following are true?[CSIR-June 2017] [duplicate]

Let $\{a_n\}$ be a sequence of real numbers satisfying $\sum _{n=1}^{\infty }\left|a_n-a_{n-1}\right|<\infty $, then the series $\sum_{n=1}^\infty a_nx^n,x\in \mathbb R$ is convergent. (a)nowhere ...
4
votes
2answers
62 views

Why compare $s_n:=\sum_{k=1}^n\frac{1}{k}$ to $\int_{1}^{n+1}\frac{1}{x}dx$, instead of $\int_1^{n}\frac1{x}dx$, when proving divergence of $s_n$?

I've seen an example in which the limit of the sequence $s_n = \sum_{k=1}^n \frac{1}{k}$ is proved to be divergent because: $$s_n \geq \int_{1}^{n+1}\frac{1}{x} dx$$ and $\log(n+1)=+\infty$ as $n\...
0
votes
0answers
45 views

Find generalized series for any value of $n\ge2$ based on given series? [closed]

If for $n=2$ we have series $0,1,3,6,10,15$ for $n=3$ we have $0,1,2,4,6,8,10$ for $n=4$ we have $0,1,2,3,5,7,9$ Based on these series, I need help to derive specific series for any value of $n$...
1
vote
2answers
51 views

Closed form of many repeated summations of n

I was looking into double summations, then I thought of repeated summations. As of now, I am having difficulty simplifying, for example $$\sum_{r=1}^8...\sum_{z=1}^y\sum_{n=1}^z\sum_{i=1}^n i$$ ...
1
vote
1answer
28 views

How to evaluate the following summation with 2 numbers below sigma [closed]

I came upon this equation in a book but I have no clue how to evaluate the sigma with $s_i=-1,1$ below it
2
votes
1answer
105 views

Find the closed form of $\quad\sum_{n=1}^\infty\frac{(-1)^nH_n^{(3)}}{n^2}$

where $H_n$ is the harmonic number and can be defined as: $H_n=1+\frac12+\frac13+...+\frac1n$ $H_n^{(3)}=1+\frac1{2^3}+\frac1{3^3}+...+\frac1{n^3}$ I managed to prove $\quad\displaystyle\sum_{n=1}^...
30
votes
4answers
887 views

Closed-form of $\sum_{n=1}^\infty\frac{(-1)^{n+1}}{n}\Psi_3(n+1)=-\int_0^1\frac{\ln(1+x)\ln^3 x}{1-x}\,dx$

Does the following series or integral have a closed-form \begin{equation} \sum_{n=1}^\infty\frac{(-1)^{n+1}}{n}\Psi_3(n+1)=-\int_0^1\frac{\ln(1+x)\ln^3 x}{1-x}\,dx \end{equation} where $\Psi_3(x)$ ...
7
votes
1answer
394 views

Does this sum go infinity? [closed]

Consider a map $F:\Bbb N\to\{\pm 1\}$ such that when $x$ is odd, $ F(x):=(-1)^{(\frac{x-1}{2})}; $ when $x$ is even, $F(x):=F(y)$, where $y$ is the odd number obtained after dividing $x$ by $2$ ...
0
votes
0answers
25 views

Convergence of the sequence $\left\{a_n\right\}$ defined by $a_{n+1}=1-\sqrt{1-a_n}$ [duplicate]

The sequence $\left\{a_n\right\}$ is defined by $a_{n+1}=1-\sqrt{1-a_n}$ for all $n\geq1$ and $a_1<1$. Prove that $\left\{a_n\right\}$ is convergent. I think it can be proved by monotone ...
8
votes
1answer
181 views

Two powerful alternating sums $\sum_{n=1}^\infty\frac{(-1)^nH_nH_n^{(2)}}{n^2}$ and $\sum_{n=1}^\infty\frac{(-1)^nH_n^3}{n^2}$

where $H_n$ is the harmonic number and can be defined as: $H_n=1+\frac12+\frac13+...+\frac1n$ $H_n^{(2)}=1+\frac1{2^2}+\frac1{3^2}+...+\frac1{n^2}$ these two sums are already solved by Cornel using ...
0
votes
3answers
70 views

Prove the divergence of $\sum_{k=0}^{\infty}{\frac{1}{10k+1}}$

Prove the divergence of $\sum_{k=0}^{\infty}{\frac{1}{10k+1}}$. By the ratio test:$$\frac{\frac{1}{10(k+1)+1}}{\frac{1}{10k}}=\frac{10k}{10(k+1)+1}=\frac{10k}{10k+11}=\frac{10k}{k(10+\frac{11}{k})}=\...
2
votes
0answers
17 views

Finite representation-infinite rings

Rings are not necessarily commutative, but associate and unital here. Recall that representation-infinite means that there are infinite non-isomorphic indecomposable modules. For a natural number $m$ ...
0
votes
2answers
46 views

On a sum of infinite series

(Second image is answer) Could somebody show how exactly the last line is derived? and why are the indices on the partial sums 1 and 3? Why not some other numbers? I really don't understand..
1
vote
2answers
260 views

Does a sequence of equicontinuous functions have a pointwise convergent subsequence? [duplicate]

Let $f_k : \mathbb R \to \mathbb R$ be an equicontinuous function sequence. If for every $k$, $f_k(0)=0$ then the sequence $\langle f_k \rangle$ has a convergent subsequence. $\langle f_k \...
1
vote
1answer
38 views

How was the series derived(logarithms)

I am revising and my past papers have short answers with no detail. My question is: How was the first series derived, and then why did they put the second series to be smaller than the first one(also ...
1
vote
4answers
68 views

Convergence of the sequence $x_n= \frac{1}{n+1}+\frac{1}{n+2}+\dots+\frac{1}{2n}$

$\left\{x_n\right\}$ is a convergent sequence where $x_n= \frac{1}{n+1}+\frac{1}{n+2}+\dots+\frac{1}{2n}$. What is $\lim_{n\rightarrow\infty}x_n?$ Here are my two approaches: Using Euler's ...
0
votes
1answer
28 views

Series with exponential upper bound

Let $K > 0$. Is it then true that there is some constant $C$ independent of $K$ such that $$\sum_{n=0}^\infty e^{-2^n K} \leq C e^{-K/C}$$ Thanks for the help!
0
votes
1answer
24 views

How to get the formula of $n$th term when it has a negative sign

So I already have the answer but I don't know the formula of getting the $n$th term. The sequence is like this: __, $4, 10, 16, 22$ I know the answer is $-2$ because the interval is $6$ but how do I ...
0
votes
1answer
37 views

Complex Taylor series on a simply connected domain

Lets say there is a simply connected domain G in $\mathbb{C}$ and there is a function f which is holomorphic on G. If we calculate the Taylor series around a point a $\in$ G we'd get something like $\...
0
votes
2answers
34 views

Why do we have $(1+\frac{c}{n}+o(\frac{1}{n}))^n \to e^c$ as $n \to \infty?$ [duplicate]

We have $(1+\frac{c}{n})^n \to e^c$ as $n \to \infty$. Why does a term in $o(\frac{1}{n})$ does not change this asymptotic?
0
votes
3answers
40 views

Partial sum of a geometric series (Calculus 3)

I have worked this problem at least 30 times and am still not getting the correct answer. Can anyone tell me where I'm wrong? Partial sum of a geometric series: enter image description here $$\sum_{...