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.

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Limit of Sequence Whose Elements Depend on Previous Elements

I was solving some exercises on limits, when I came across this problem: The point $C_1$ divides a segment $AB = l$ in half; the point $C_2$ divides a segment $AC_1$ in half; the point $C_3$ ...
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Convergence of the infinite product of $\cos(1/\sqrt{i})$

I am trying to prove that the following product is convergent and show what it converges to: $$\prod_{i=1}^\infty \cos{ \left( \frac{1}{\sqrt{i}} \right)}$$ I have heard that products are convergent ...
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1answer
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Another interesting task connected with De Bruijn sequence

Rule "zero is better than one": Consider a sequence which is made up only of elements from the set {0, 1} and constructed regarding to the following rules: 1) It starts with n ones. 2) We put one ...
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Uniform convergence of $\sqrt{x^2+n^2}^{-1}-n^{-1}$

Assume the following sum: $$ \sum_{n=1}^{\infty} f_n(x) = \sum_{n=1}^{\infty}\left(\frac{1}{\sqrt{x^2+n^2}}-\frac{1}{n}\right), $$ where $x \in \mathbb{R}$. The problem is to determine whether the sum ...
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84 views

Let $a>0$, show that $\sum(1+a^n)^{-1}$ is divergent if $0<a\leq1$ and convergent if $a>1$

Let $a>0$, show that $\sum(1+a^n)^{-1}$ is divergent if $0<a\leq1$ and convergent if $a>1.$ What I did: If $0<a\leq1$, then $(1+a^n)^{-1}\geq 1/2$, so the series is divergent. Similarly ...
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1answer
39 views

Can you compare sequences with information about their series?

Let $a_n$ and $b_n$ be a sequence of non-negative, non-increasing numbers such that $$A(x):=\sum_{n\leq x} a_n \leq \sum_{n\leq x} b_n := B(x)$$ for all $x$. Can you conclude that $a_n \ll b_n$? ...
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2answers
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Sequence Convergence proof check

I am trying to solve the following sequence question: Show that $\{a_n\}_{n=1}^\infty$ converges to $A$ if and only if $\{a_n - A\}_{n=1}^\infty$ converges to $0$. Proof: Assume $\{a_n\}_{n=1}^\...
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1answer
146 views

Example that $\lim \sup (x_n\cdot y_n)<\lim \sup (x_n)\cdot \lim \sup (y_n)$

This is a short question, I already managed to prove using definitions that $$\lim \sup (x_n\cdot y_n)\le \lim \sup (x_n)\cdot \lim \sup (y_n)$$ But I'm having trouble coming up with an example such ...
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Detect five consecutive unsorted integers

It appears that the product of the differences between 3 consecutive integers in whatever order is always equal to 2. However, I can't find the pattern for 5 integers. We could sort them and get an ...
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1answer
38 views

Proving convergence of series $\sum_{n=1}^{\infty}u_{n}$

I was asked to prove that the series; $\sum_{n=1}^{\infty}u_{n}$ converges, given that, $\lim_{n \rightarrow \infty}n^{p}u_{n}=A<\infty$ and $p>1$. The proof of this is: If $u_{n}<\frac{A}{...
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152 views

Statements possibly equivalent to the $\epsilon{-}N$ definition of limit of a sequence

Which of these statements are equivalent to the $\epsilon{-}N$ definition of limit of a sequence? (a) For every integer $m>0$, there in an integer $N>0$ such that if $n > N$ then $|...
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2answers
57 views

Proving that the point spectrum of the right shift operator on $\mathscr{l}^2(\mathbb{Z})$ is empty.

How can I prove that no series $x\in\mathscr{l}^2(\mathbb{Z})$ of the form $$\forall i\in\mathbb{Z}: x_{k-1}=\lambda x_k$$ exists other than the zero sequence? In particular I want to prove that the ...
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1answer
64 views

Is there a theorem relating sequences to its series or vice versa?

I only have these in mind Theorem: If a series $\sum_n a_n$ of real numbers converges then $\lim_\limits{n \to \infty} |a_n|=0$ Divergence test: If $\lim_\limits{n \to \infty} a_n \neq 0$, then the ...
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1answer
82 views

Condition for derivative sequence to converge?

Let $E_n(R)$ be a sequence of function that converge to $E(R)$, When we can said that $\dot{E}_n(R)$ converge to $\dot{E}(R)$. I can assume: uniform converge of $E_n$ to $E$. $E(R)$ convex. ...
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1answer
54 views

To what value will the series converge?

I have done a Fourier series expansion and get $$\frac{12}{\pi(2n-1)}\sin((2n-1)x)$$ How to find the value it converges at $x=\frac{\pi}{2}$? isn't it divergent? Please show me the correct way step by ...
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2answers
85 views

A summation with binomial coefficients

Evaluate There seemed to be some problem with stackexchange's math rendering but Ian corrected whatever error was there in the expression.Thanks $$5050 \frac {\left( \sum _{r=0}^{100} \frac {{100\...
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267 views

mean square convergence vs almost sure convergence

I saw a few examples that show that almost sure convergence doesn't imply convergence in mean square. Can anyone find an example of a random series that converges in mean square but doesn't converge ...
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1answer
34 views

I need to show that this sequence is increasing and I'm almost there but I need help on last step.

Let $(1+\frac{1}{n})^n$ be a sequence and $f(x)=(1+\frac{1}{x})^x $ on $[1,inf)$. I need to show that f is non-decreasing by showing that $f'(x)\ge0$. So far I have: Let $g(x)=ln(f(x))$, where $ln$ ...
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1answer
98 views

upper bound of some weird function with an exponencial

well I'm trying to find the proof of the following statement, but I cant go forward anymore: Let $\Omega= \mathbb{R}^{n} \times (0,\infty) $ $$w(x,t)= \sum^{\infty}_{k=0} \frac{1}{(2k)!}\frac{d^kg(t)}...
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1answer
209 views

If $3x^{2}-2(a-d)x+(a^{2}+2(b^{2}+c^{2})+d^{2})=2(ab+bc+cd)$, then

If $3x^{2}-2(a-d)x+(a^{2}+2(b^{2}+c^{2})+d^{2})=2(ab+bc+cd)$, then $A.$ a,b,c,d are in G.P. $B.$ a,b,c,d are in H.P. $C.$ a,b,c,d are in A.P. $D.$ None of the above Tried writing the expression as a ...
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1answer
191 views

Determining if a series is absolutely convergent or conditionally convergent without the usage of the limit comparison test

Would the following series be conditionally convergent, absolutely convergent or divergent? $$\sum^\infty_{k=1}\frac{k\sin{(1+k^3)}}{k+\ln{k}}$$ Whereas for sine functions in series like this, you ...
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2answers
265 views

Is the sequence $\{(-1)^n 1/n\}$ divergent?

Is the sequence $\{(-1)^n{1\over n}\}$ divergent or convergent? It's convergent to $0$ because for every $\varepsilon$, exists $1/N < \varepsilon$ by archimedean property, so $|1/n|< \...
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0answers
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Application in geometric sequence

A wire has a total length of 8184cm. It was cut into various pieces to form a series of 10 squares. The length of each subsequent square doubles of the previous square. If the length of the first ...
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0answers
23 views

Irreducible Chaotic Polynomials

I'm working to generate discrete chaotic sequences - directly in an algebraic field. In a reference: Digital Chaotic Communications by Alan Michaels https://smartech.gatech.edu/bitstream/handle/...
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1answer
33 views

Problems using the sum of geometric series

$2^0 + 2^1 + 2^ 2 + 2^3+...+2^{n+1}$ According to the general formula, the above sequence can be summed by $\frac{r^{n+1}-1} {r-1}$. If I plug the parameters from the above sequence I don't get the ...
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1answer
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How to find the smallest $N$ in tolerance questions in sequences and series?

Question : Let the GP be $1,\left (\frac{-2}{3}\right), {\left (\frac{-2}{3}\right)}^2,{\left (\frac{-2}{3}\right)}^3, ... $ Choose a numerical tolerance $\epsilon=0.0005$. Determine the smallest ...
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1answer
69 views

Does this series have a region of convergence?

$$\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\cdots}}}}$$ I was working on this series from the UKMT website. I solved it to get the two roots. (1 + sqr5)/2 (Golden ratio number). Now I am asking myself what ...
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1answer
35 views

How to solve recurrence relations involving integrals?

Suppose we have two sequences $C_i$ and $f_i$, where $C_0=\frac{\pi^2}{3}$ and $f_0=i\pi x$ and with the recurrences $$f_{n+1}(x)=\int \left( \int ( f_n(x) + C_n ) dx \right)dx$$ where the integrals ...
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2answers
32 views

Uniform Convergence of $\sum_{n=1}^{\infty}\frac{x}{[(n-1)x+1][nx+1]}$

What can be said about the uniform Convergence of $\sum_{n=1}^{\infty}\frac{x}{[(n-1)x+1][nx+1]}$ in the interval $[0,1]$? The sequence inside the summation bracket doesn't seem to yield to root or ...
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1answer
66 views

Show that $(1+\frac{1}{n})^n=\sum_{k=0}^{n}\frac{1}{k!}\Rightarrow \lim_{n\rightarrow\infty}(1+\frac{1}{n})^n=\sum_{k=0}^{\infty}\frac{1}{k!}=:e$

I write out the left term expression and get $$ \sum_{k=0}^{n}\binom{n}{k}\bigg(\frac{1}{n}\bigg)^k $$ If I could Show that the k-th term of both sequences is equal I would be done. I.e what I want ...
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1answer
27 views

If $a$, $b$, $c$ form a geometric progression, then $\sqrt{a}$, $\sqrt{b}$, $\sqrt{c}$ also form a geometric progression.

I'm trying to teach myself some A-level maths (in the UK) from a text book, but have come unstuck with the following question: If $a$, $b$, and $c$ are the first three terms of a geometric ...
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1answer
44 views

Proving that equality is applicable to any N number

$a_{n}$ is a sequence of numbers defined by: $a_{0}$ = −1 $a_{1}$ = 3 $a_{n+2}$ = 6$a_{n}$ − $a_{n+1}$ + 4n + 1 I have to prove that for every natural number n this equality is applicable: $...
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2answers
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Series, limits and convergence

Can you help me with it with working $\sum _{n=1}^{\infty }\:\cos \left(\frac{2n+1}{n^2+n}\right)\sin \left(\frac{1}{n^2+n}\right)$ I am struck in the part where i get, $\lim _{n\to \infty \:\:}\...
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3answers
47 views

Real number sequences and limits.

Could anyone help me? I have to solve this problem, justifying it: $\{a_n\}$ and $\{b_n\}$ are sequences of real numbers. Given that: $\displaystyle\lim_{n\rightarrow\infty} a_n$ = a > 0 and $\...
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2answers
55 views

Show that $\sum_{n=1}^{\infty}\frac{\log (1+1/n)}{n}$ converges

I need some help here. I can show that if the Cauchy condensation test holds, then I get two separate series, one which converges by the comparison test, and one that converges by the ratio test. But ...
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1answer
45 views

How to get Taylor Series of $\sin \frac{x}{1-x}$

I know that $\displaystyle\sin x = \sum_{k=0}^{\infty}\frac{\left( -1 \right) ^kx^{2k+1}}{(2k+1)!}$ But how to get transformation to get Series about x?
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1answer
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Partition of the index set of the non-negative series

Suppose that $a_j\geq 0$ for all $j\in \mathbb{N}$. Prove that $$\sum \limits_{j\in A}a_j+\sum \limits_{j\in B}a_j\leq \sum \limits_{j=1}^{\infty}a_j,$$ where $A$ and $B$ - partition of the natural ...
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1answer
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Help finding formula for infinite series

I am looking for the correct way to write the following infinite series using summation notation. It is similar to a geometric series, just with certain terms missing. $\frac{1}{2} + \frac{1}{2^3} + \...
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1answer
62 views

Can $ \displaystyle \sum_{n=1}^{\infty} \frac{2 + (-1)^n }{1.25^n} $ be split into two?

Can $$\sum_{n=1}^{\infty} \frac{2 + (-1)^n }{1.25^n} $$ be split into two so that it could be solved without the comparison test? I am thinking of splitting the sum into to: the first one will be a ...
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2answers
40 views

series $\frac{3}{1}+\frac{4}{2^p}+\frac{5}{3^p}+\frac{6}{4^p}+\cdot\cdot\cdot$

For what p series:$$\frac{3}{1}+\frac{4}{2^p}+\frac{5}{3^p}+\frac{6}{4^p}+\cdot\cdot\cdot$$ is convergent? Answer: $\frac{3}{1}+\frac{4}{2^p}+\frac{5}{3^p}+\frac{6}{4^p}+\cdot\cdot\cdot=\frac{1+2}{...
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1answer
36 views

Formula to calculate sum of amount at 7% rate to 10th term.

I'm developing a Unity Game and I want to calculate below thing fast. 4, 4.28, 4.58, 4.90 Next number is 7% addition to the earlier number. 4 * 7 / 100 + 4 = 4.28; 4.9 - 4.58 = .32; 4.58 - 4....
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2answers
34 views

real infinite series

Let $p>1$. Prove that $$\sum^{\infty}_{n=1} \frac {\sqrt[p] {n^{p-1}}}{(n+1)n} <\infty.$$ I got that the numerator is less than $n$ so that I want to find something that is lower than the ...
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1answer
24 views

generating sequence with a specific recursive condition

I'm looking for two different sequences that satisfy $x_n=x_{2n}+x_{2n+1}$ for all $n\geq 1$, whats a general possible way to find such sequences/ what other sequences satisfy that condition? my idea:...
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1answer
48 views

Lights on game algorithm

When I was a young child I was on holiday with my mother with a dutch family. One day Dick, the family man, handed me a wooden box with some switches and lights (maybe eight, but I'm not sure) and ...
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1answer
65 views

How to prove this problem about sequences

A sequence of positive integers $(a_n)_{n\ge1}$ contains each positive integer exactly once. Assume that if $m\neq n$ then $$\frac1{1998}<\frac{\left|a_m-a_n\right|}{|m-n|}<1998.$$ Prove that $\...
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3answers
65 views

Does this cubic sequence converge or diverge?

Prove that the sequence $$a_n = 8n^3 + n^2 - 2$$ either converges or diverges. If it converges, find the value it converges to. What I have so far: Since the $$\lim_{ n\rightarrow \infty} a_n= \...
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3answers
55 views

Need help to compute the limit of a series

So I have the following exercise: exercise photo I have determined that the series is convergent. But I do not know how to compute the limit.
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1answer
79 views

How to find the finite or infinite sum of a given series?

Consider $k$ with $\frac 12≤ k < 1$ Define the set $A = \{k^n\}_{n=1}^{\infty}$ How can we prove that if $x\in (0,1)$ then there is a finite or infinite sum of the numbers in A (repetitions not ...
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1answer
55 views

partial sum of the series involving trigonometric function

Let $U_r=cos(\theta +(r-1)a)$ and I have to find $f_{(r)}$ such that, $2sin(\frac{\alpha}{2})U_r=f_{(r+1)}-f_{(r)}$ I was managed to find $f_{(r)}$ such that $2sin(\frac{\alpha}{2})U_r=f_{(r+1)}-f_{...
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166 views

How to investigate convernegce of $\sum_{n=1}^{+\infty} \frac{(2x)^n}{x^2n+\frac{70}{n}}$ for $x \in \mathbb{R}$

I need to investigate convergence and absolute convergence of series $$\sum_{n=1}^{+\infty} \frac{(2x)^n}{x^2n+\frac{70}{n}}$$ depending on the value of the parameter $x \in \mathbb{R}$ I think use ...