# 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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### Assume $x_n>0$,$x_n+\dfrac{4}{x_{n+1}^2}<3.$ Prove $\lim\limits_{n \to \infty}x_n$ exists and evaluate it.

My Solution Notice that $$x_n+\dfrac{4}{x_{n+1}^2}<3=3\sqrt[3]{\frac{x_n}{2}\cdot\frac{x_n}{2}\cdot\frac{4}{x_n^2}}\leq \frac{x_n}{2}+\frac{x_n}{2}+\frac{4}{x_n^2}=x_n+\frac{4}{x_n^2}.$$ This ...
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### Finding a general formula for the sequence {$0, 1, 3, 5, 6, 7, 9, 15$}.

I'm working on a mathematical model which requires me to generalize the elements of an infinite set. The element n is the nth finite sequence of the set. For: ...
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### Weird sum that is almost definitely not $\sqrt 2$

I have not the ability to compute more than four digits of $$\sum_{n=1}^\infty \frac{1}{n^2 H_n^{(\ln n)}}$$ $H_n^{(m)} = \sum_{k=1}^n \frac{1}{k^m}$ is the generalized harmonic number. I know ...
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### Series whose Cauchy product is absolutely convergent - A general example

Is there series that is divergent or conditionally convergent with absolutely convergent Cauchy product? Seems like there is a group of these examples! Perhaps finding divergent series with ...
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### Does there exist a series with property $n\sum_{k=0}^{\infty} u_{nk}=1$?

Shortly, the idea is to find such series which admits "lazy" calculation: instead of computing all the terms, it would be enough to calculate its even terms (the case with $n=2$), and then multiply ...
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### Disproving the existence of a specific infinite sequence of Fibonacci primes

Consider the following sequence: $$T_{1} = a,\: T_{i+1} = F_{T_{i}}$$ where $a \in \mathbb{P}$ and $F_i$ is the $i$-th Fibonacci number. Is there a value of $a \neq 5$ such that this sequence ...
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### Pointwise limit of continuous functions, but not Riemann integrable.

I am trying to find a simple example of a function $f:[0,1]\rightarrow\mathbb{R}$ which is a pointwise limit of continuous functions, but is not Riemann integrable. I know the classical example where ...
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### Possible formula for $f(x) = \sum_{n=0}^{\infty}x^{-n!}$

I was wondering if we have a formula for the following function: $$f(x) = \frac{1}{x^{0!}} + \frac{1}{x^{1!}} + \frac{1}{x^{2!}} + \frac{1}{x^{3!}} + ... = \sum_{n=0}^{\infty}x^{-n!}$$ (Like we ...
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### Limit at Infinity of Maclaurin Series

Let $f(x) = \sum_{n=1}^\infty a_n x^n$. What is $\lim_{x\rightarrow \infty} f(x)$ in terms of the $a_i$? That question may be too broad, so here are some restrictions: Assume f(x) is continuous (and ...
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### Find an explicit formula for the recursive formula

Find an explicit formula for the recursive formula: $$a_{n+1} = 2a_n\left(a_n + 3\right); a_0 = 4$$ The first few terms in the sequence go like this: $4, 56, 6608, \dots$ After $a_2$ the sequence ...
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### Consider the metric space of infinite sequences of 0s and 1s under this metric.

For $x, y ∈ \{0, 1\}^\mathbb{N}$, define $d(x, y) = 2^{-n}$ where $n$ is the first position where the sequences $x$ and $y$ are different. Show that ($\{0,1\}^\mathbb{N}, d$) is compact. Show that ...
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### Finding invariant measures (to solve recurrences)

INTRO: If you don't feel like reading the justification for my question, just skip down to the question. Some recurrence relations can be solved by finding invariants, or quantities that remain ...
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### The fractal dimension of the Kolakoski sequence is $2-1/e$

The Kolakoski sequence, which is defined as the infinite sequence of symbols {1,2} that is its own run-length encoding (Wikipedia), has been suggested to be self-similar$^{1}$. The fractral ...
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### Alernative way of solving summation $5+55+555+\ldots$

Question Find the summation for the series-: $$5+55+555+5555+...$$ I know it is a duplicate of this, but still, I am posting this because i was thinking of solving another way due to which I got ...
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### Closed form for $\sum_{n=1}^\infty \left(e-\left(1+\frac{1}{n}\right)^n \right)^2$?

Is there a closed form for this series: $$\sum_{n=1}^\infty \left(e-\left(1+\frac{1}{n}\right)^n \right)^2 \approx 1.273278374727530507449$$ (Mathematica computation by Patrick Stevens). This ...
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### Limit points of a sequence constructed from pi (if pi is normal)

I'm interested in this question apparently posed by John Nash, which I found in the book A Beautiful Mind. If you make up a bunch of fractions of pi $3.141592\ldots$. If you start from the ...
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### Finding a quadruplet of “cow-equivalent” integers

This a follow up question/puzzle from an earlier question here How does rounding affect Fibonacci-ish sequences? Explanation For this new question, one needs only to understand a particular ...
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### A generalization of the Euler-Mascheroni constant

Let $f:[1,+\infty)\rightarrow \mathbb{R}$ be a differentiable function. We are dealing with the limit of the sequence $$f(n)-\sum_{k=1}^nf'(k).$$ If $f=\log$, then it is convergent to $-\gamma$ (...
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