For questions about recurrence relations, convergence tests, and identifying sequences

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0answers
31 views

Find the general term of $a_{n+1} = \sqrt{2a_n + 3}$

Find the general term of $a_{n+1} = \sqrt{2a_n + 3}$, where $a_1 = 1$. Is this possible?
0
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2answers
20 views

Limit of a sequence problem

Suppose that $(x_n)$ is a sequence such that $\lim_{n\to\infty}\sum_{k=1}^n\frac{{x^4_k}}n=0.$ How do I show that $\lim_{n\to\infty}\sum_{k=1}^n\frac{{x_k}}n=0$?
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2answers
104 views

Is the sum of positive divergent series always divergent?

If two positive terms series $\sum_{n=1}^{\infty} a_n, \sum_{n=1}^{\infty} b_n$ are divergent, $\sum_{n=1}^{\infty} (a_n+b_n)$ is also divergent. I thought is was obvious, but I saw a counterexample ...
1
vote
1answer
67 views

Convergence of series of the form $\sum 1/n^x$

Why the solution says $\sum_{n=1}^\infty \dfrac{1}{n^{1.5}}$ converges? Does every series $\sum_{n=1}^\infty \dfrac{1}{n^{x}}$ converges to 0 except $1/n$ (harmonic Series)? I found that after ...
3
votes
1answer
22 views

Calculating a bound on the norm of a matrix exponential

The problem is this: Let A be a square $n \times n$ matrix, and define $$e^A=\sum_{k=0}^\infty \frac{1}{k!}A^k$$ Find a bound for $\lvert e^A \rvert$ in terms of $\lvert A \rvert$ and $n$. I was ...
2
votes
1answer
79 views

Finding the sum of this Gamma series

I am trying to compute the sum of the following series $$\sum _{k=0}^{\infty }\frac{\left(2it (1-H)^{2 (1-H)} \left(\frac{H}{\mu}\right)^{2 H} \right)^k \Gamma \left(\frac{k}{2 (1-H)}+\frac{1}{2 ...
1
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1answer
40 views

$0\leq a_{n}-l\leq \dfrac{\pi^{2} }{2^{2n+1}}$

this is related to that one the limits of $a_n $and $b_n$ Let for $n\geq 2\quad a_{n}=\prod\limits_{k=2}^{n}\cos\left(\dfrac{\pi }{2^{k}}\right)$ and $b_{n}=a_{n}\cos\left(\dfrac{\pi ...
1
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2answers
32 views

the limits of $a_n $and $b_n$

this is related to that one $a_n$ is bounded and decreasing Let for $n\geq 2\quad a_{n}=\prod\limits_{k=2}^{n}\cos\left(\dfrac{\pi }{2^{k}}\right)$ and $b_{n}=a_{n}\cos\left(\dfrac{\pi ...
2
votes
3answers
49 views

$a_n$ is bounded and decreasing

my second question from [An inequality for the product $\prod_{k=2}^{n}\cos\frac{\pi }{2^{k}}$ Let $n\geq 2\quad a_{n}=\prod\limits_{k=2}^{n}\cos\left(\dfrac{\pi }{2^{k}}\right)$ and ...
4
votes
3answers
189 views

Help with evaluating this (likely telescopic) summation.

I am trying to solve this problem which is regarding evaluating summation: $$\sum_{k=1}^{\infty}\frac{6^{k}}{(3^{k}-2^{k})(3^{k+1}-2^{k+1})}$$ Points to note: It seems to be telescopic summation ...
1
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3answers
41 views

$b_{n}$ is increasing

I think there is misunderstanding in my last post because its contain three questions so i will post question by question step by step An inequality for the product $\prod_{k=2}^{n}\cos\frac{\pi ...
21
votes
3answers
205 views

How to prove that $\frac{\zeta(2) }{2}+\frac{\zeta (4)}{2^3}+\frac{\zeta (6)}{2^5}+\frac{\zeta (8)}{2^7}+\cdots=1$?

How can one prove this identity? $$\frac{\zeta(2) }{2}+\frac{\zeta (4)}{2^3}+\frac{\zeta (6)}{2^5}+\frac{\zeta (8)}{2^7}+\cdots=1$$ There is a formula for $\zeta$ values at even integers, but ...
1
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1answer
34 views

An inequality for the product $\prod_{k=2}^{n}\cos\frac{\pi }{2^{k}}$

I have this hint from old question of mine if someone could help me to understand it Sequence $0\leq a_{n}-l\leq \dfrac{\pi^{2} }{2^{2n+1}}$ for the first question it's easy to see that a_n is ...
1
vote
1answer
38 views

Pattern on polynomials disguising as exponentials

Recently I've been looking at integer sequences that look like exponential at the first few terms but is actual polynomial, like these two sequences [1] [2]. And there seems to be something ...
9
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2answers
193 views

Equivalence of $\pi$ is the first positive zero of the taylor series for $\sin(x)$ and $\pi/4 = 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \cdots$

For $x\in\mathbb{R}$, define $\sin (x) = x - x^3/3!+x^5/5!-\cdots$ and $\pi = 4(1-\frac{1}{3}+\frac{1}{5} -\frac{1}{7}+\cdots)$. Then show that $\sin(\pi/2) = 1$ In the prologue of Real and Complex ...
1
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3answers
47 views

Sequence and series problem

How do I show that the sequence $(x_n)$ defined by $x_ {n+1} = \left(1-\frac{1}{n}\right) ^2 x_n + \frac{1}{n}, \forall n \in \Bbb{N}$ converges? and to what limit?
0
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1answer
31 views

nth Partial Sum $S(n) = \frac{n - 1}{n + 1}$

Given: Let: $$S = \sum^{\infty}_{n=1} a(n)$$ be an infinite series such that the nth partial sum is given by: $$S(n) = \frac{(n - 1)}{(n + 1)}$$ Find $a(3)$ $a(1):$ ...
0
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2answers
21 views

Probability after repetition and series

So I was wondering about how often you would have to try something for a certain probability if you had a 15% winrate. Then I found something weird: The average loss rate l with n tries is obviously ...
1
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0answers
41 views

Planning to integrate $\int_{0}^{\infty} \frac{\log^2(x)}{x^2 + 1} dx$ using complex analysis [duplicate]

This is just a plan-out. I want to evaluate: $$\int_{0}^{\infty} \frac{\log^2(x)}{x^2 + 1} dx$$ Using a keyhole contour a semi-circle, with base at the x-axis. First I must pick a branch. ...
1
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1answer
38 views

solution verification

here is solution of my old question but i can't see it would someone explain to me the principal idea and what he wants to show Solution from ...
23
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2answers
161 views

Generalisation of $(n+3)^2-(n+2)^2-(n+1)^2+n^2=4$

After seeing the neat little identity $(n+3)^2-(n+2)^2-(n+1)^2+n^2=4$ somewhere on MSE, I tried generalising this to higher consecutive powers in the form $\sum_{k=0}^a\epsilon_k(n+k)^p=C$, where $C$ ...
2
votes
1answer
22 views

Probability of a sequence of characters within a random sequence of characters

I've been trying to work out a formula for the probability of a specified sequence of upper-case English characters of length $n$ appearing in a random sequence of upper-case English characters of ...
1
vote
2answers
34 views

Series of non-zero rational terms converging to a rational number?

On a recent test, I was asked if the following statement was true or false: Every convergent series of non-zero rational terms converges to an irrational number I marked it as false because I don't ...
0
votes
2answers
58 views

How to solve specific infinite series

I have two questions in particular that have been bothering me. Any help would be greatly appreciated: Consider the sum $$\sum_{n=1}^\infty n^{2}x^n$$ Determine the interval in which it is ...
0
votes
0answers
40 views

Show this equality is equal [duplicate]

![show this equality is equal][1] the equality is $$\frac{(1+2x+x^2) (1+x)}{(1-2x+x^2)(1-x)} = 1 + [ 6x + 18x^2 + 38x^3 + 66x^4 +\ldots ]$$
6
votes
5answers
1k views

Formula to convert 1 to 5… to 5 to 1

Is there a single formula to convert the following? And would it be based on reciprocals? 1 to 5 2 to 4 3 to 3 4 to 2 5 to 1 Ideally, I'd like to convert 0 to 5 1 to 4 2 to 3 3 to 2 4 to 1 but ...
0
votes
1answer
43 views

Find the limit of right angles ( is it possible?) [on hold]

My question is it possible to find the limit of right angles which is what the question seem to be asking?
2
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2answers
85 views

Binomial theorem.

I first saw this thing (admittedly much to late in life) in a third year class entitled non-linear dynamics and chaos theory. There if i am remembering correctly we used to look for non-zero terms to ...
4
votes
1answer
52 views

Is $f(0) + \sum_{n=1}^\infty f(n) - f(n-1)$ equal to $\lim_{x\to\infty} f(x)$ or $0$?

If I have a sum for which I remove the previous iteration every time, then I am left with only the last iteration of the sum: $$f(0) + \sum_{n=1}^x f(n) - f(n-1) = f(x)$$ If this would be a sum to ...
2
votes
1answer
42 views

Show complex sequence is convergent

We have complex sequence $a_n$ such that $\displaystyle \sum_{n=1}^{\infty}a_n$ is convergent. Let $\sigma : \mathbb{N} \to \mathbb{N}$ be a bijection where we know that there exist $M \in \mathbb{N}$ ...
0
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6answers
72 views

Convergence of a sequence given by $x_{n+1}=\frac 23(x_n+1)$

A sequence $(x_n)_{n\in\mathbb N}$ is defined by $$x_0=0, \qquad x_{n+1}=\frac23(x_n+1)\text{ for }n=1,2,\dots$$ Prove that this sequence converges and find the limit. The infimum of $(x_n)$ ...
-1
votes
1answer
46 views

can't show directly from the definition that is not Cauchy sequence [on hold]

I need to show, directly from the definition, that each of the following is not Cauchy sequence. Please help, because I haven't been able to make any progress. a: $\;( n^2 )$ b: ...
1
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2answers
69 views

A series for Fibonacci numbers.

How can I prove The Fibonacci sequence is encoded in the number $1/89$ i.e. $( 1/89 = 0.01 + 0.001 + 0.0002 + 0.00003 + 0.000005 + 0.0000008 + 0.00000013 + 0.000000021 + 0.0000000034 \ldots)$
0
votes
3answers
75 views

$\int_{0}^{1} (x-1)\sqrt{1-x} dx$ without Parts etc..

Evaluate: $$\int_{0}^{1} (x-1)\sqrt{1-x} dx$$ Without the use of integration by parts (1) My initial thought is, can we use series for either one of these? Can we find a series (about x=0) for ...
1
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1answer
36 views

How to determine if this sequence converges or diverges?

How to determine if this sequence converges or diverges? if it converge can someone show me, but if it diverges how can we know all the sub-limit of it? ...
0
votes
0answers
49 views

Inequality regarding sequence approximation square root

I was reading a proof in which the square root function on $[0,1]$ is approximated by a sequence of functions (polynomials) defined on $[0,1]$. The induction hypothesis is that you have functions ...
5
votes
1answer
66 views

Is this series $\sum_{n \geq 2}\sqrt{a_n}\frac{n^{a_n}-1}{\ln n}$ always divergent?

Let $\displaystyle \sum a_n$ be a series with positive terms which is a convergent series and suppose that we don't have $\displaystyle a_n \ln n \rightarrow 0$. Is the following series always ...
1
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0answers
17 views

Is this a viable generalization of Newton series?

I wonder if the following formula a viable generalization of Newton's series. $$f(x)=\frac{1}{2\pi}\int_{-\infty}^{+\infty} (- i \omega)^x \int_{-\infty}^{+\infty}e^{i\omega t}\sum_{m=0}^\infty ...
1
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1answer
35 views

convergent subsequences etc.

It is well known that: A sequence of real numbers converges $\{a_n\}$ to p, if and only if every subsequence $\{a_{n_k}\}$ converges to p. I am wondering if this similar statement holds.: ...
3
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1answer
51 views

Estimating the divergence of a “convex series.”

An exercise from R.C. Buck's Advanced Calculus: Let $f≥0$, $f'≥0$, $f'' \geq 0$ for $1≤x<\infty$. Show that $$0≤ \sum_1^n f(k) - \int_1^n f(x)dx - \frac12f(n) - \frac12f(1)≤\frac14f'(n)$$ for ...
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2answers
74 views

Let $\{a_n\}_{n=1}^\infty$ be an infinite sequence. Does there exist an infinite series whose partial sums is $\{a_n\}_{n=1}^\infty$?

Definition: By an Infinite Sequence of real numbers, we shall mean any real valued function whose domain is the set of all positive integers. Definition: By an Infinite Series of real numbers, we ...
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0answers
72 views

Sequences problem from AMM [on hold]

A cute problem I think, but the (official) solutions are somewhat unnatural. Would be interesting in seeing some alternative approaches. Suppose $x_1,x_2,...$ is a sequence of positive real numbers ...
1
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3answers
58 views

Induction on nth polynomial proof.

Question: Prove by induction that $ 1+r+r^2+\cdots+r^n = \dfrac {1-r^{n+1}} {1-r} $ where $ r \in \mathbb{R} $ When $n$ is odd, this is really easy as the right side breaks down to $\dfrac ...
0
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0answers
37 views

trouble with sequence and series question [on hold]

$x^2f''(x)+xf'(x)+(x^2-1)f(x)=0$ show that $f(x) = \displaystyle\sum_{n=0}^{\infty} \dfrac{(-1)^n}{n!(n+1)!2^{2n+1}}x^{2n+1}$ Dont know to approach this one
1
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5answers
146 views

Power series for the rational function $(1+x)^3/(1-x)^3$

Show that $$\dfrac{(1+x)^3}{(1-x)^3} =1 + \displaystyle\sum_{n=1}^{\infty} (4n^2+2)x^n$$ I tried with the partial frationaising the expression that gives me $\dfrac{-6}{(x-1)} - ...
0
votes
2answers
37 views

Showing that the $\lim s_n\neq\dfrac{2}{3}$ when $s_n=\dfrac{2}{3n}$

I'm trying to verify if I what I did to show that limit does not exist is valid using the negation of the definition: $\exists \epsilon>0, \forall N \in \mathbb{N}$ such that $n>N \implies ...
-1
votes
0answers
58 views

Sequence with Conditions and Possible Answers [on hold]

Given that $a_1$, $a_2$, $a_3$, . . . $a_n$ is a sequence of positive real numbers such that: For all positive integers $m$ and $n$, $a_{mn}$ = $a_m$$a_n$, AND there exists a positive real number $B$ ...
3
votes
5answers
73 views

Proving the convergence of $\sum_{n\geq 2}\log\left(1-\frac{1}{n^2}\right)$

How to prove that the series $$\sum_{n\geq 2}\log\left(1-\frac{1}{n^2}\right)$$ is convergent? What about finding the sum? My attempt: $$\ln (1-1/n^2)= \ln(n-1) -2\ln n + \ln(n+1)$$ The term in the ...
-5
votes
0answers
29 views

Finding the sum [on hold]

How to find the sum of $\sum_{n=1}^{\infty}\frac{(-1)^n}{2^nn}$ ?