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

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0
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3answers
25 views

Showing that this sequence is eventually decreasing

I'm trying to show that this sequence $$a_n = \frac{3^n-7}{4^n+5}$$ is decreasing for all $n$ greater than some $N\in \Bbb N$. All I can see to do is something like $$a_{n+1} = ...
0
votes
1answer
26 views

Find the range of $x$ for which the sequence $\dfrac{n!} {k!(n-k)!}x^n $ converges to $0$ for a stabilised $k\in\mathbb{N}$

I'm studying in preparation for a Mathematical Analysis I examination and I'm solving past exam exercises. If it's any indicator of difficulty, the exercise is Exercise 1 of 4, part $c$ and graded for ...
4
votes
3answers
91 views

Limit of $a_{n+2}=a^2_{n+1}+\frac{1}{6}\cdot a_n+\frac{1}{9}$

Find a limit of sequence: $$a_{n+2}=a^2_{n+1}+\frac{1}{6}\cdot a_n+\frac{1}{9}$$ $$a_1=0,a_2=0$$ I tried to prove that $a_n$ is bounded and monotonic, but I couldn't prove that $a_n$ is monotonic (by ...
0
votes
1answer
24 views

Is there any identity for this series?

While solving inequality and finite series problem I often come across this series- $$(n+1)(n+2)(n+3)...(n+n)$$. Is there a general solution to this form of a series? Thanks for any help!!
0
votes
0answers
15 views

Can you define a Cauchy product from this identity?

I can write vague expessions for $\zeta(3)$, the Apéry's constant, for example when I multipliy by $\frac{1}{n^4}$ the recursion relation (2) in page 2 here, and after I take the sum ...
1
vote
0answers
27 views

Closed form of an integral $\int_0^{\pi/2} \ln^n (\sin x) \, dx$

Let $n \in \mathbb{N}$. May we have a closed form for the integral: $$\mathcal{J}=\int_0^{\pi/2} \ln^n (\sin x) \, {\rm d}x$$ One obvious approach would be to go through beta functions and ...
2
votes
2answers
30 views

Can you get a closed-form for $\sum_{j=0}^{\infty}\frac{2^{2j-1}B_{2j}}{(2j)!}$?

Let $B_{k}$ the kth Bernoulli number, then using their asymptotic I can justify the absolute convergence of this series $$\sum_{j=0}^{\infty}\frac{2^{2j-1}B_{2j}}{(2j)!},$$ since, if there are no ...
1
vote
3answers
28 views

What is the general term of the sequence $u_{n+1}=c u_n+d$? [on hold]

What is the general term of the sequence $u_{n+1}=c u_n+d$ ?
1
vote
1answer
19 views

Error with the proof that all solutions to the Cauchy Functional Equation are linear

If $f(x)$ is continuous, it is known that $f(x+y)=f(x)+f(y)$ implies that $f(x)$ is linear, and non-continuous solutions are discussed in these links. (1, 2,3, 4) However, what is wrong with this ...
2
votes
2answers
56 views

Does $n\pi - \lfloor n\pi\rfloor$ have a subsequence that goes to zero?

Does $n\pi - \lfloor n\pi\rfloor$ have a subsequence that goes to zero? I was talking to a friend about it but neither of us were able to come up with anything. We're not sure if $\pi$ is essential ...
0
votes
3answers
32 views

Arc length of a sequence of semicircles.

Let $\gamma_1$ be the semicircle above the x axis joining $-1$ and $1$. Now divide this interval into $[-1,0]$ and $[0,1]$, and trace a semicircle joining $-1,0$ above the $x$-axis and another one ...
0
votes
2answers
22 views

Using the definition of the limit sequence, find $\lim_{n\rightarrow\infty}\frac{n}{n^2-2}$ for $n=2,3,4,…$

Using the definition of the limit sequence, find $\lim_{n\rightarrow\infty}\frac{n}{n^2-2}$ for $n=2,3,4,...$ I've tried to isolate n but its impossible to do.
0
votes
1answer
27 views

Bounded away sequence implications

Consider the sequence $\{\sqrt{n}|a_n-a|\}_n$ where $a_n, a \in \mathbb{R}$. Assume $\{\sqrt{n}|a_n-a|\}_n$ is bounded away from $0$ and $\infty$. Is this equivalent to or sufficient or necessary for ...
1
vote
1answer
13 views

Implications of $n a_n=O((n^{\frac{1}{\alpha}}|b_n|)^\alpha)$

Consider two sequences of real numbers $\{a_n\}_n$, $\{b_n\}_n$. Suppose $n a_n=O((n^{\frac{1}{\alpha}}|b_n|)^\alpha)$ where $\alpha \in \mathbb{R}$, $na_n\geq 0$ and big $O$ notation is explained ...
-2
votes
1answer
32 views

How to get from left to right-hand side of the equation? $ \sum_{k=0}^{d} \binom{2d+1}{k} = \frac{1}{2} \cdot 2^{2d+1} $

I would like to know how the left hand side of the equation is achieved. In particular why the $\frac{1}{2}$ is there. I don't understand how one can get from the left to the right side. $$ ...
1
vote
2answers
26 views

Cluster points and the sequence 1,1,2,1,2,3,1,2,3,4,1,…

I am working on a problem in analysis. We are given a sequence $x_n$ of real numbers. Then a definition: A point $c \in \mathbb{R}\cup{\{\infty, -\infty}\}$ is a cluster point of $x_n$ if there is a ...
0
votes
1answer
9 views

Geometric progression with reverse order

I have the following problem: Find three positive numbers which have the sum of $70$ and create a Geometric progression ($q>0$, increasing). Their inverse sum equals to $4/70$. Thank you!
2
votes
1answer
61 views

The closed form of $\int^\infty_{B}e^{-(x+\frac{A}{x})}\,dx$, where $A>0$, $B>0$.

What tools, ways would you propose for getting the closed form of this integral? $$\int^\infty_{B}e^{-\left(x+\frac{A}{x}\right)}\,dx,$$ where $A>0$, $B>0$. When $B=0$, from Table of ...
4
votes
2answers
38 views

Sequence bounded away from $0$ and $2$

Suppose I have a sequence of real numbers $\{a_n\}_n$ and I'm told that $\{a_n\}_n$ is bounded away from $0$ and $2$. (1) What does it mean exactly? My thinking is that it means $a_n\neq 0$ and $a_n ...
1
vote
1answer
36 views

Proving non-repitition of a sequence

I have heard that the sequence $$x_{n+1}=rx_n(1-x_n)$$ for $r$ between $3$ and $4$ does not recur i.e. there is no $a>0$ such that $x_{n+a} = x_n$ and $x_0$ is any number between 0 and 1 exclusive. ...
0
votes
0answers
30 views

all but one sub-strings within a cyclic string

over $GF(q)$ where $q\in\mathbb{N}$, we build a string of size $q^n-1$. now, how can I show that it is impossible to construct that string so it contains all sub-strings of size $n$ exactly once, but ...
1
vote
0answers
44 views

Estimate from above the series: $\sum_{i=1}^n \frac{1}{(i-j)^2}$.

Estimate from above the series: $$\sum_{i=1}^n \frac{1}{(i-j)^2}$$ for $n\in\mathbb N$, $j=1,2,\dots,n$ and $j\neq i$. I'd like to know an estimate dependent by $j$. This question comes from a simple ...
1
vote
0answers
19 views

Denominators of harmonic numbers: asymptotic behaviour.

About the sequence $d_n$ of the denominators of harmonic numbers, I know these facts: It is unbounded, since $p\mid d_p$ for any prime $p$. It contains only one $1$. What more is known? Specially, ...
0
votes
1answer
32 views

Given series diverges or converges?

Given: $$\sum_{i=1}^\infty \frac2{7*i + 21} $$ The limit of the $nth$ term is 0, it means we aren't sure if it diverges. On wolfram it says it diverges by comparison test, but how?
0
votes
1answer
19 views

Calculus Series and simplifying the expression

Considering the following series, Series How do I find a simplified expression for the ratio (an+1 / an)
0
votes
1answer
47 views

Maximum value of $\lambda$

It is given that a,b,c are be of same sign and a,b,c are in Harmonic progression i.e. $\frac{2}{b}=\frac{1}{a}+\frac{1}{c}$ and also $\frac{a+b}{2a-b}+\frac{c+b}{2c-b}\geq \sqrt{\lambda \sqrt{\lambda ...
3
votes
1answer
50 views

Prove if $\displaystyle \sum_{n=1}^ \infty a_n$ converges, {$b_n$} is bounded & monotone, then $\displaystyle \sum_{n=1}^ \infty a_nb_n$ converges.

Prove that if $\displaystyle \sum_{n=1}^ \infty a_n$ converges, and {$b_n$} is bounded and monotone, then $\displaystyle \sum_{n=1}^ \infty a_nb_n$ converges. No, $a_n, b_n$ are not necessarily ...
0
votes
3answers
20 views

Prove the existence of limit of certain sequences.

Problem: Let $0<a_1<b_1$ and $$a_{n+1}=\sqrt{a_n\cdot b_n},b_{n+1}=\frac{a_n+b_n}{2}.$$ Prove that $\{a_n\}$ and $\{b_n\}$ converge to some limit. Attempt: By induction and AM-GM, I can show ...
0
votes
0answers
10 views

Infinite series, continued fractions, nested radicals and such - is there a general recursive algorithm theory?

The nature of infinity and irrational numbers (among other things) always gave me trouble. But recently I learned to think about infinite sequences in terms of recursive algorithms and their time ...
-2
votes
2answers
62 views

Cant find the pattern [on hold]

These are two sequences: I tried looking for a ratio or difference but it wasnt working $$ 12,20,32,\underline{\quad},22,\underline{\quad},50 $$ $$ 6,10,16,8,\underline{\quad},22\underline{\quad} $$ ...
0
votes
0answers
35 views

Show that $\lim_{n\rightarrow \infty} (1-\omega(\frac{1}{n}))^n=0$ and $\lim_{n\rightarrow \infty} (1-o(\frac{1}{n}))^n=1$

Could you help me to show that (1) $\lim_{n\rightarrow \infty} (1-\omega(\frac{1}{n}))^n=0$ (2) $\lim_{n\rightarrow \infty} (1-o(\frac{1}{n}))^n=1$ where $o(\cdot)$ is little $o$ notation described ...
-2
votes
0answers
41 views

What is the sum of the series: $1+2(1/3)+3(1/3)^2+4(1/3)^3+\cdots$ [duplicate]

Please help me find the sum of this series. I tried separating the geometric series pattern from it but the remaining part is just the same.
0
votes
1answer
22 views

Limit of a function defined as the sum of a series

Given a decreasing sequence $a_n$ of positive real numbers, for $x>0$ define $$ f(x)=\sum_n \min\left(\frac{x}{a_n}, \frac{a_n}{x}\right). $$ Can $a_n$ be chosen so that $f(x)\to 0$ as $x\to 0$?
0
votes
1answer
29 views

Convergence of the sequence $z_1=\frac{3}{2}$ with $z_{n+1}=\sqrt{3z_n-2}$

Prove the convergence of the sequence $(z_n)$ such that : $$ z_1=\frac{3}{2}$$ $$z_{n}=\sqrt{3z_{n-1}-2}$$ for every $ n \geq 2$. Calculate also the limit. I have applied induction: $$\frac{3}{2} ...
5
votes
7answers
134 views

Convergence of the recursive sequence $a_{n+1}=\sqrt{2-a_n}$

I'm attempting to prove the convergence (or divergence, though I strongly suspect it converges) of a sequence defined as $a_{n+1}=\sqrt{2-a_n}$ with $a_1=\sqrt{2}$. I cannot use the monotonic ...
-1
votes
0answers
48 views

Proof of 1^2 + 2^2 + 3^2 + 4+2 +… infinity = 0 [on hold]

Leonhard Euler proved that the sum of the series $$ 1^n + 2^n + 3^n + 4^n + ... ∞ = 0 $$ where n is an even number. I know that this is a diverging series, but the infinity part of it somehow makes ...
2
votes
1answer
62 views

Evaluate $\lim_{x \to \infty} \frac{(\frac x n)^x e^{-x}}{(x-2)!}$

$$\lim_{x \to \infty} \frac{(\frac x n)^x e^{-x}}{(x-2)!}$$ where $x$ is $\mathbb N$-valued and $n$ is some nonzero real number. Wolfram seems to give $0$ for different values of $n$ that I tried. ...
0
votes
2answers
27 views

find the limits of…

$$A_n=\frac{cos(n)}{n}\to 0$$ I've been asked to find the limits of this sequence however using the sandwich theorem I have just found that the limit is $0$ is this the only limit if not what else is ...
0
votes
0answers
10 views

Smoothing a function by subtracting terms in its Taylor series?

I am looking at some code for a Greens function that mentions the following % The GF is the smoothed by subtraction of first two odd Taylor series terms. So how does subtracting terms from a ...
2
votes
2answers
46 views

Solving the recurrence $A_n = \sum_{k=1}^{n} 2^{k+1} A_{n-k}$

Let me ask a very simple question: Let $(A_n)$ be a sequence of integers defined by $A_0 = 1$ and $$\forall n \geq 1 : A_n = \sum_{k=1}^{n} 2^{k+1} \cdot A_{n-k}.$$ There is an explicit formula for ...
0
votes
2answers
30 views

How do I evaluate this sum :$\sum_{n=1}^{\infty}\frac{{(-1)}^{n²}}{{(i\pi)}^{n}}$?

I'm interesting to know how do i evaluate this sum :$$\sum_{n=1}^{\infty}\frac{{(-1)}^{n²}}{{(i\pi)}^{n}}$$, I have tried to evaluate it using two partial sum for odd integer $n$ and even integer $n$ ...
0
votes
2answers
29 views

What do we mean by convergence of a series?

While learning calculus I stumbled upon this concept of convergence. Is this some general concept or just related to sequence and series. What is its importance?
1
vote
4answers
73 views

can't determine the convergence/divergence here

Let $$t_{n}=\frac{1}{n}\left(1+\frac{1}{\sqrt{2}}+\dots+\frac{1}{\sqrt{n}}\right),\ n=1,2,\dots$$ then I want to know if $\sum_{n=1}^{\infty}t_{n}$ converges/diverges and the sequence$\{t_{n}\}$ ...
0
votes
1answer
24 views

For a sequence, why must $\lim _{n→∞} {||x_n||} = ∞$, $\lim _{n→∞} {||x_n||} = 0$, or there exists a convergent subsequence with a nonzero limit?

Suppose I've got a sequence of vectors $\{x_n\}_{n∈N}$ in $\mathbb{R}^k$. Why is it that exactly one of the following three facts must hold: $\lim _{n→∞} {||x_n||} = ∞$, $\lim _{n→∞} {x_n} = 0$, or ...
2
votes
2answers
69 views

Compute $(\sin4^\circ)^2 +(\sin8^\circ)^2+(\sin12^\circ)^2+\cdots+(\sin176^\circ)^2$

Angle of sine is in degrees, can anyone show me an easy soln to this? This was question was given to us for 1minute without calcu. I know that $\sin4^\circ=\sin176^\circ$, ...
2
votes
0answers
31 views

For x between 0 and 1, what does the following series converge to?

Consider the following infinite series: $\sum_{i = 1}^{\infty} 2^i x^{(2^i)} = 2x^2 + 4x^4 + 8x^8 + 16x^{16} + ...$ I know that it converges for values of x between 0 and 1- can anyone help find a ...
1
vote
0answers
14 views

Ratios between index and sequence element

Let $a_1\le a_2\leq\ldots$ be an infinite sequence of positive integers. A positive integer $n$ is called good if $i=na_i$ for some index $i$. For which $(m,n)$ is it true that if $m$ is good, then ...
3
votes
1answer
67 views

Is there a name for an infinite product series?

I already know about the Harmonic series: $$\sum_{n = 1}^{\infty} \frac 1n = 1 + \frac 12 + \frac 13 + \frac 14 + \frac 15 + \frac 16 + \cdot \cdot \cdot$$ But is there a name for this infinite ...
1
vote
0answers
9 views

Monotonically decreasing sequence/series proof [duplicate]

I have a proof that I'm working on and it goes like this: Assume $a_k > 0$ and $a_k$ is monotonically decreasing. Show that: $$\sum_{k=1}^\infty a_k < \infty \iff \sum_{k=0}^\infty b_k < ...
0
votes
2answers
24 views

Question concerning limit superior and inferior.

Let ${a_n}$ and ${b_n}$ be two real sequences such that $a_n\leq b_n$ for all $n$. Is it true that $\lim \sup a_n \leq \lim\inf b_n $? Outline the proof if so.