# Questions tagged [sequences-and-series]

For questions concerning sequences and series. Typical questions concern, but are not limited to: identifying sequences, identifying terms, recurrence relations, $\epsilon-N$ proofs of convergence, convergence tests, finding closed forms for sums. For questions on finite sums, use the (summation) tag instead.

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### Generalized of Miklos Schweitzer 1980 P1

question; For a real number $x$, let $\|x \|$ denote the distance between $x$ and the closest integer. Let $0 \leq x_n <1 \; (n=1,2,\ldots)\ ,$ and let $\varepsilon >0$. Show that there ...
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### Problem with the convergence region of the series

Can the region of convergence of a functional series defined on the real axis consist of a half-interval and a segment? Yes maybe! I understood this intuitively, but I couldn’t show it clearly. I ...
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### $\inf_{n\in \mathbb{N}^*} \left\{\frac{3^n}{2^n} \right\} > 0$?

The fractional part is defined of a positive real number $x$ is defined by $\{x\}:=x-\lfloor x\rfloor$. Is it true that $$\inf_{n\in \mathbb{N}^*} \left\{\frac{3^n}{2^n} \right\} > 0 \ ?$$ If so, ...
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### Application of Bertrand's postulate [duplicate]

We can use Bertrand's Conjecture ( that for any integer $n \not= 0$, there exists at least one prime number $𝑝$ with $n < p \leq 2n$ ) to demonstrate the ...
1 vote
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### 2 concise tables of “usual” series (mostly trigonometrics) and of "usual" L-series (Zeta, Eta, Beta...)

CONTEXT Common series are usually described as infinite sums, written as consecutive terms ending with (…). Or they can be described using the $\sum_{}$ symbol and arguments usually including $(-1)^k$ ...
1 vote
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### Boundedness of the sequence $\left\{\sum_{k=1}^n \frac{1}{k}\sin\frac{n}{k}\right\}$.

I want to study the boundedness of the sequence $\left\{\sum_{k=1}^n \frac{1}{k}\sin\frac{n}{k}\right\}$. In one of my old post $\lim_{n\to\infty}\sum_{k=1}^n \frac{1}{k}\sin\frac{n}{k}$, we got that: ...
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### Real analysis supremum of a set [closed]

Define supremum of a set $S\subset R$ and show that the $\sup S =\frac{1}{2}$ where $S=\{\frac{n}{2n+1} \mid n\in N\}$.
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1 vote
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### An example application where convergence in probability is useful or valuable

I have been learning different notions of convergence for sequences of random variables. I know the almost sure convergence is the strongest. The next best thing is convergence in probability. Can ...
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### Need help with generalized binomial formula (a+b)^n where neQ [closed]

Something is really bugging me about the binomial expansion with rational exponents. Maybe someone can explain this to the non-math person that I am. Say we are interested in the first 4 terms in the ...
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### Definition and Use of the Schett Polynomial in the Jacobi Taylor Series

I am having a tough time understanding the definition and use of the Schett polynomial introduced in the paper here. I have two questions related to this polynomial. My first question concerns its ...
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1 vote
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### Showing sequence converges knowing its bounded and $a_{n+1} - a_n \geq-\frac1{2^n}$ for all $n$ [closed]

Let $(a_n)$ be bounded sequence, s.t for all $n \in \mathbb{N}$, $a_{n+1}\geq a_{n} - \dfrac{1}{2^n}$, I'm stuck on showing $(a_n)$ converges.
1 vote
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### Multiplication of convergent sequence and multiplication of series

It is a basic rule that $((\lim_{n\to \infty}a_{n}=A\in\mathbb{R})\wedge (lim_{n\to \infty }b_{n}=B\in\mathbb{R})\Rightarrow lim_{n\to\infty}(a_{n}b_{n})=AB)$ And according to the definition of ...
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### What is wrong with this proof of the Bolzano-Weiserstrass theorem?

I tried proving the Bolzano-Weierstrass theorem before looking at the solution in my math textbook. This is what I tried: Take a bounded sequence $x_n$. Every sequence has a monotone subsequence ...
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### Formula for Root Series

Given the series: $S_n=\sqrt{c}+\sqrt{c-1}+\sqrt{c-2}+\sqrt{c-3}+...+\sqrt{c-n}$ in which $c$ can be any value but independent from n. Is there a formula for these kinds of series? Excluding the ...