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Results tagged with sequences-and-series
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user 494920
for questions about sequences and series, e.g. convergence, closed form expressions, etc. Note that there is a different tag for spectral sequences, and also note that MathOverflow is not for homework. Please consider consulting the online encyclopedia for integer sequences, if you are trying to identify a given sequence that you have found in your research.
6
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0
answers
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Tweak the numerators in the alternating harmonic series so that the partial sums alternate a...
I was thinking about the alternating harmonic series: $1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\cdots$.
I wondered what would happen if we tweak the numerators so that the partial sums alternate between …
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25
votes
3
answers
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Does there exist a continuous function $f(x)$ such that $f(0)=0$ and $0<\lim_{n\to\infty}\pr...
Does there exist a continuous function $f(x)$ such that $f(0)=0$ and $0<\lim\limits_{n\to\infty}\prod\limits_{k=1}^n f(\frac{k}{n})<\infty$ ?
I do not see any reason why such a function could not exis …
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31
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5
answers
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On average, how many uniformly random real numbers $u$ are needed for their sum to exceed $1...
A well-known question is: on average, how many uniformly random real numbers in $(0,1)$ are needed for their sum to exceed $1$? The answer is $e$.
Let's tweak this question by making each random numbe …
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0
votes
You have $n$ rectangles of area $1$ and variable height. Pack as many as possible in a semic...
I have evidence that, for large $n$, the number of leftover rectangles is approximately $\color{red}{\frac16\log{n}+c}$, where $c\approx 0.482$.
We will pack the circles from $\color{red}{\text{top to …
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15
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0
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387
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Will a unit disk be completely covered by randomly placed disks of area $\pi,\frac{\pi}{2},\...
On a "bottom" disk of area $\pi$, we place "top" disks of area $\pi,\frac{\pi}{2},\frac{\pi}{3},\dots$ such that the centre of each top disk is an independent uniformly random point on the bottom disk …
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0
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1
answer
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If $a_1=1$ and $a_n=\sec (a_{n-1})$ then what does the proportion of positive terms approach...
Consider the sequence $a_1=1$ and $a_n=\sec (a_{n-1})$ for $n>1$.
What does the proportion of positive terms approach, as $n\to\infty$?
At first I thought the limiting proportion might be $\frac{1} …
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9
votes
1
answer
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You have $n$ rectangles of area $1$ and variable height. Pack as many as possible in a semic...
You have $n$ rectangles of area $1$ and variable height. Pack as many of these rectangles as possible in a semicircle of area $n$. How many leftover rectangles will there be, in terms of $n$?
How to p …
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