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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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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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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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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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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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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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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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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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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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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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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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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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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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 ...