For questions about recurrence relations, convergence tests, and identifying sequences. For questions on finite sums use the (summation) instead.

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2
votes
2answers
58 views

Help me understand this sequence problem

Today, I encountered a problem in "Problem-Solving Strategies" by Arthur Engel (Chapter $9$. Sequences, page-$225$): Prove that there does not exist a monotonically increasing sequence of ...
1
vote
1answer
58 views

Finding a convergent majorant series

I have a series $$ \sum_{n=1}^\infty \left( \frac{1}{n^3} \cos(nt) - \frac{1}{(2n+1)^2} \sin(nt) \right) $$ and I have to find a majorant series to this series. The convergent majorant series I was ...
9
votes
1answer
161 views

Closed form for Euler sum with $H_{2n}$?.

I ran across this Euler sum while trying to evaluate an integral. I mentioned it in another thread, but though perhaps asking about it separate may be a good idea. Is there a closed form for this ...
1
vote
8answers
131 views

Prove that $( 1 + n^{-2}) ^n \to 1$.

I need to prove that $\left(1 + \frac 1 {n^2} \right)^n \to 1$. I tried to use Bernoulli's inequality, but that is not very useful since in the original sequence there is a plus sign. I then tried to ...
0
votes
0answers
20 views

Find convergent majorant series

I have a series $\sum_{n=1}^\infty \frac{1}{n^3} (\cos(nt) - 2 \sin(nt))$ and I want to find a convergent majorant series. I know that I have to find $k_n$ such that $|f_n(t)| = |\frac{1}{n^3} (\cos(...
3
votes
1answer
82 views

sum, least upper bound of infinite series

I don't know how to find the sum (not as a decimal number) or the least upper bound of the infinite series $$\sum_{{k=1}}^{\infty} \frac{(k!)^2}{(k^2)!}$$
5
votes
2answers
133 views

Sum (least upper bound) of infinite series

I have to find sum (or the least upper bound) of infinite series (exact expression not decimal number) of series $\begin{equation} \sum_{k=1}^{+\infty}\frac{k!(1+k)^k}{(k^2)!} \end{equation}$. I am ...
1
vote
0answers
48 views

Congruence of Euler numbers modulo Fermat numbers

The exponential generating function of the unsigned Euler numbers $E_{n}$ is $$\frac{1}{\cos x}=\sum_{n\ge 0}\frac{ E_n}{n!}x^n$$ For $k,n\gt0$, not too large, one can observe that $$E_{ 2^{2^k}\...
0
votes
1answer
21 views

M compact. $(x_n)$ converges if and only if $(x_n)$ has only one closure point [duplicate]

M compact. $(x_n)$ converges if and only if $(x_n)$ has only one closure point. Show, with one example, that the compacity is necessary. My attempt: 1) Of course if $(x_n)$ converges, $(x_n)$ has ...
1
vote
1answer
51 views

Finding a continuous $f\in C(\mathbb{T})$ with $\{\hat{f}(n)\}\notin \ell^p$

Denote the circle by $\mathbb{T}=\mathbb{R}/\mathbb{Z}$. Find an example for a continuous function $f\in C(\mathbb{T})$ with coefficients of the fourier series, $\{\hat{f}(n)\}\notin\ell^p,\forall p&...
3
votes
2answers
106 views

The coefficient of $x^{15}$ in the product $(1-x)(1-2x)(1-2^2x)(1-2^3x)\cdots (1-2^{15}x)$ is …

Problem : The coefficient of $x^{15}$ in the product $(1-x)(1-2x)(1-2^2x)(1-2^3x)\cdots (1-2^{15}x)$ is (a) $2^{105}-2^{121}$ (b) $2^{121}-2^{105}$ (c) $2^{120}-2^{104}$ (d) $2^{110}-2^{108}...
1
vote
1answer
44 views

Expression for $1 - 2^z x + 3^z x^2 - 4^z x^3 + \cdots$

Using Taylor series we have $$\frac 1 {(1+x)^2} = 1 - 2x + 3x^2 - 4x^3 + \cdots$$ Then multiplying by $x$ and differentiating we get $$\frac {1-x} {(1+x)^3} = 1 - 4 x + 9 x^2 - 16 x^3 + \cdots$$ ...
2
votes
2answers
43 views

Properly Divergent Sequences

Using the definition in Bartle's Introduction to Real Analysis, I am trying to gain an intuitive understanding of limits that tend to infinity. Given Definition: Let ($x_n$) be a sequence of real ...
5
votes
1answer
95 views

Question regarding $f(n)=\cot^2\left(\frac\pi n\right)+\cot^2\left(\frac{2\pi}n\right)+\cdots+\cot^2\left(\frac{(n-1)\pi}n\right)$

$$f(n)=\cot^2\left(\frac\pi n\right)+\cot^2\left(\frac{2\pi}n\right)+\cdots+\cot^2\left(\frac{(n-1)\pi}n\right)$$ then how to find limit of $\dfrac{3f(n)}{(n+1)(n+2)}$ as $n$ tends to infinity? I don'...
1
vote
0answers
30 views

Multisection of a Power Series Proof

Suppose that $$H_{N,k}(x)=\frac{x^ke^{\frac{-x}{N}}}{N^{k-1}k!\sum_{n=0}^{N-1}{w_N^{-nk}e^{\frac{w_N^nx}{N}}}}=\sum_{n=0}^\infty{A_n\frac{x^n}{n!}}$$ where $k\lt N, w_N=e^{\frac{2i\pi}{N}}$, and ...
0
votes
0answers
35 views

the series of a function applied to every summand of a convergent series converges only if the function is a constant factor.

I want to show the following: Let $\varphi: \mathbb R \to \mathbb R$ a function with the following property: For every convergent series $\sum^\infty_{n=1} x_n$ with $x_n \in \mathbb R$ for all $n \...
2
votes
2answers
31 views

Given $(a_n)$ with $a_n \geq a_{n+1} \geq 0$ show that if $\sum \sqrt{a_n a_{n+1}} < \infty \implies \sum a_n < \infty$

I'm having some difficulty solving this problem : Given the sequence $(a_n)$ with $a_n \geq a_{n+1} \geq 0$ show that if $$\sum \sqrt{a_n a_{n+1}} < \infty \implies \sum a_n < \infty.$$ I ...
3
votes
1answer
111 views

Why is this set compact in $L^2(\mathbb{N})$?

Suppose $L^{2}(\mathbb{N})$ is the Hilbert space of sequences $(a_{n})_{n \in \mathbb N}$ which satisfy $\sum |a_{n}|^{2}$ with $(a,b) = \sum a_{n} \bar{b_{n}}.$ Prove the set of sequences $\{a_{n}\}...
0
votes
0answers
24 views

If $x_{k,n}=a+\frac{k(b-a)}n$ for $k=1,2,3,…,n$, then $\lim_{n\to\infty}\frac1n \sum_{k=1}^n f(x_{k,n})=\frac1{b-a}\int_a^bf(x)dx$

Assume that f is integrable over $[a,b]$ If $x_{k,n}=a+\frac{k(b-a)}n$ for $k=1,2,3,...,n$, then $\lim_{n\to\infty}\frac1n \sum_{k=1}^n f(x_{k,n})=\frac1{b-a}\int_a^bf(x)dx$ My Work: I like to ...
0
votes
3answers
114 views

Limit of the Sequence: $a_{n+1}=a_n+\frac{1}{a_n}$ with $a_1=1$

$\lim_{x\to\infty} a_n$ defined as $a_{n+1}=a_n+\frac{1}{a_n}$ with $a_1=1$. I've tried writing out the first few terms and this sum is increasing, but progressively less-so over time. I'm just not ...
0
votes
0answers
24 views

How to prove an equation involving $\sum_{p,n}\frac{\log (p)}{p^{n/2}}\delta_{\log (p^n)}(y)$

My context is that I could assume as true statements (I say this since claims about the type of convergence could be difficult to me) the first equation in [1], that could be written as the derivative ...
0
votes
2answers
24 views

Trouble finding a sequence formula

The sequence goes: 1,$\frac{2}{3}$,$\frac{7}{9}$,$\frac{20}{27}$,$\frac{61}{81}$ I tried using the common difference method of analysis and found the second row follows the rule: multiply by $\frac{-...
1
vote
2answers
42 views

Finding the limit of the following series, wherever it is convergent

Let $ f(x) = \sum_{n=1}^{\infty}a_nx^n $ where $a_n$ is the nth Fibonacci number. Find the values for which the series is convergent, and find $f(x)$ for those values. (i.e - find the limit when ...
0
votes
0answers
34 views

Is there a name for this property (on sequences)

Suppose we have a sequence $(x_n)_{n=1}^\infty\subseteq X$ with $(X,d)$ a metric space, and we have the following property: $$\exists x\in X\forall\epsilon>0[|B_\epsilon(x)\cap (x_n)_{n=1}^\...
2
votes
1answer
33 views

Intersection of three period functions

Let $f(x)=\frac{1}{2}-|\frac{\sqrt{3}}{2}x-1/2|$ for $x\in [0,\frac{2}{\sqrt{3}}]$ and $f(x+\frac{2}{\sqrt{3}})=f(x)$ for all $x\in\mathbb{R}$. $g(x)=\frac{1}{2}-|\frac{1}{2}x-1/2|$ for $x\in [0,2]$ ...
1
vote
1answer
61 views

Is this series convergent? (which could not be solved by comparison theorems)

Let $$c_n=\sum_{k=1}^{2n-1}\frac{1}{k^a}\left(\frac{1}{(2n-k)^a}-\frac{1}{(2n+1-k)^a}\right)$$ Can we show that $\sum_{n=1}^\infty c_n$ converge? If $a>1$, then by Kay K., it is OK to be ...
2
votes
4answers
182 views

limit of a sequence defined recursively, is my proof correct?

So i have a sequence defined by $a_1 =1$ and $a_{n+1} = a_n + \frac{1}{a_n}$ and i would like to know $\lim_{n\rightarrow \infty} a_n$. I have said that the sequence $a_n$ is unbounded and thus the ...
2
votes
2answers
57 views

Let $\{a_n\} ,\{b_n\}$ be given bounded sequences of positive real numbers.

Let $\{a_n\} ,\{b_n\}$ be given bounded sequences of positive real numbers. Then (Here $ a_n\uparrow a$ means $a_n$ increase to a n goes to $\infty$, similarly, $b_n\downarrow b$ means $b_n$ decreases ...
2
votes
1answer
96 views

Laurent Series for $\csc(z)$

I have to find the Laurent series for $$\csc(z), \qquad |z|>0 $$ but I really don't know how to start. Please, guys.
1
vote
1answer
58 views

Is this number in $O(\log(n))$?

Is this number $\big[\log(n) + \sum_{j=1}^{n-1} (\log(j) - (j+1)(\log(j+1)) + j \log(j) +1)\big] \in O(\log(n))$? I simplified it to $\big[\log(n) + \sum_{j=1}^n (-\log(n+1) - j(\log(n)) + 1)\big]$.
4
votes
1answer
105 views

Generalization on older SE question: not just covering $1$, but all rational numbers.

Older SE question lies here. So I will change the question such that you can understand the question better: $$\sum_{c\space\subset \Bbb{Co}}\frac{1}{c}\le k$$ -where your goal is to get to $k$ ...
1
vote
2answers
55 views

Convergence of a sum of series

I was reading the famous "Calculus" by Spivak and at the beginning of the chapter on infinite series, he states: "It's an easy exercise to prove that if both $\displaystyle \sum_{k=1}^{\infty}a_n$ and ...
0
votes
2answers
105 views

showing that the partial sums of $ \log(j) = n\log(n) - n + \text{O}(\log(n))$

I'm trying to show that the partial sums of $\log(j) = n\log(n) - n + \text{O}(\log(n))$ I know that $$\int_1^n\log(x)dx = n\log(n) - n + 1$$ so that this number is pretty close to what I want. Now ...
5
votes
1answer
69 views

Limit of a specific sequence involving Fibonacci numbers.

Let, $\left\{F_n\right\}_{n=1}^\infty$ be the Fibonacci sequence, i.e, $F_1=1, F_2=1~\&~ F_{n+2}=F_{n+1}+F_n~\forall ~n \in \mathbb{Z}_+$ Let, $P_1=0, P_2=1$. Divide the line segment $\overline{...
3
votes
2answers
83 views

Convergence of $\sum_{n=1}^{\infty}f(n)$ $\iff $ convergence of $\int_1^{\infty}$f(x)dx

f is a real valued $C^1$ function on [0,$\infty$]. Suppose that $\int_1^{\infty}|f'(x)|dx$ converges. Show that Convergence of $\sum_{n=1}^{\infty}f(n)$ $\iff $ convergence of $\int_1^{\infty}$f(x)dx. ...
2
votes
1answer
43 views

Interchange of $\ell^r$ and $L^p$-norm

Let $(f_i)_{i\in\mathbb{N}}$ be a sequence of $L^p$-functions. What is the relation between $\Vert \Vert (f_i)_{i\in\mathbb{N}}\Vert_{\ell^r}\Vert_{L^p}$ and $\Vert \left(\Vert f_i\Vert_{L^p}\right)_{...
1
vote
0answers
28 views

tight bound for a finite sum involving harmonic series

I want to a know tight bound of this quantity when $n$ is even $$\sum_{k=1}^{n/2}\sum_{m=n-k}^{n}\frac{1}{k(k+1)m}$$. I simplified the expression and it comes like $$H_n[1-\frac{1}{n/2+1}]-\sum_{k=1}...
4
votes
2answers
174 views

Sum of a Series With Denominators of the form $(2^i) (3^j)(5^k)$

Can anyone solve this? Find the sum of the series $1 + \frac{1}{2} +\frac{1}{3} + \frac{1}{4} + \frac{1}{5}+ \frac{1}{6} + \frac{1}{8} + \frac{1}{9} + \cdots,$ where the denominators are of the form ...
3
votes
0answers
37 views

Proving a Particular Coefficient of a Power Series Equals $0$

Suppose I have a particular function $$F(x,z) = \sum_{n=0}^\infty{A_n(x)\frac{z^n}{n!}}$$ and suppose, through the use of a particular computer algebra system, that the particular polynomial ...
0
votes
1answer
46 views

A question on summation of series…

If $$\frac{\left(1^4+\frac14\right)\left(3^4+\frac14\right)\ldots\left((2n-1)^4+\frac14\right)}{\left(2^4+\frac14\right)\left(4^4+\frac14\right)\ldots\left((2n)^4+\frac14\right)}=\frac1{k_1n^2+k_2n+...
1
vote
1answer
37 views

Convergence of a series involving Riemann Zeta Function

I was wondering how to show whether this series converges or not: $\sum_{m=0}^{\infty}(-1)^{m}\frac{\pi^{2m+2}}{(2m+2)!}\zeta(-2m-1)$ Numerically it converges after a few terms in wolfra alpha. But ...
5
votes
2answers
642 views

Help me evaluate this infinite sum

I have the following problem: For any positive integer n, let $\langle n \rangle$ denote the integer nearest to $\sqrt n$. (a) Given a positive integer $k$, describe all positive integers $n$ such ...
1
vote
3answers
58 views

Explain why series divergent or convergent

See question title. The series is as follows: $$\sum_{n=1}^\infty{2n+\sin{n}\over {e^n-\cos n}}$$ Now, common sense dictates that the numerator "goes to infinity" much slower than the denominator, ...
1
vote
0answers
27 views

Is this definition correct regarding the logic of recursive sequences?

This is a self study. I come across with a second order sequence of reals defined by: $$φ_{m+2}=φ_{⌊f(φ_{m},φ_{m+1})⌋}..................(*)$$ where $⌊.⌋$ is the integer part function. Here $f$ is a ...
0
votes
1answer
30 views

determine convergence or divergence of a series, finding sum of series

I am given the series $$\sum_{n=2}^\infty \frac{1+2^n}{3^n}$$ So I used ratio test for this one to see if this converges and I found out that it converges, however the answer key tells me there is a ...
0
votes
2answers
52 views

Prove divergence of sequence

$a_n=1+n\sin\left(\frac{n\pi}{2}\right)$ So the above sequence is obviously not convergent because of the sine, but how could one prove this? Thanks for any pointers. Tom
1
vote
1answer
68 views

Does this series of functions converge?

Let $g(x)$ be a real valued differentiable function on $\mathbb{R}$ with $g(0)=0$, and $|g'(x)|\leq M$ for all $x \in \mathbb{R}$. Let $$\displaystyle f(x) = \sum_{n=1}^{\infty} \frac{1}{n} g\bigg(\...
0
votes
2answers
91 views

Number of strings lenght $n$ with no consecutive zeros

I know this problem is familiar to "the number of binary strings of length $n$ with no consecutive zeros" but still it is a way different problem. But in that question the alphabet is set to $\{0,1\}$ ...
2
votes
4answers
27 views

Testing infinite series for convergence [closed]

I have the following series $\sum_{n=1}^\infty ne^{-n^2}$ How would one go about showing convergence? Which test can be used?
3
votes
6answers
99 views

Limit of $n\ln\left(\frac{n-1}{n+1}\right)$

I'm having trouble with this limit: $$\lim_{n\to ∞}n\ln\left(\frac{n-1}{n+1}\right)$$ It's supposed to be solvable without using l'Hospital's rule. I'm guessing it's a case for the squeeze theorem, ...