Questions tagged [upper-lower-bounds]

For questions about finding upper or lower bounds for functions (discrete or continuous).

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How small is $\sum_{d \mid p_n\#}\mu(d)\sum_{r^2 = 1 \pmod{p_{n+1}d}}\frac{(x - r) \pmod {p_{n +1}d} + 1}{p_{n+1}d}$?

Rough Conjecture: Define $f(x) = \sum_{d \mid \sqrt{x + 1}\#}\mu(d)\sum_{r^2 = 1 \pmod d} \frac{(x - r) \pmod d}{d}$ where the modulus operation takes the least non-negative residue in $\Bbb{Z}$. ...
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bounding the tail of $p$-series with trig factor

The tail of $p$-series ($p > 1$) can be upper bounded by (from e.g. here): $$\sum_{j=n}^\infty \frac{1}{j^p} = O(n^{1-p}).$$ I'm interested in finding the upper bound of ...
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finding a tight scaling bound (in terms of the Big-O notation) of a function of an infinite sum of $1/n^2$.

I have a real-valued function $f(x)=\sum_{n=x+1}^\infty \frac{1}{n^2}$ where $x \in \mathbb{N}$. I want to understand how $f(x)$ scales with respect to $x$. One thing I tried is as follows: Since it ...
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Upper bound for the expectation of a random variable times an indicator function

Suppose we have a random variable $X$. Is there any way of deriving an upper bound of the following expectation: $$E[X * \mathbf{1}_{X\ge x_0}],$$ where $\mathbf{1}_{()}$ is an 0-1 indicator function, ...
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Minimum number of edges for a tree that joins the $27$ nodes of a $3 \times 3 \times 3$ regular grid

In 2014, Dumitrescu and Tóth (see Covering Grids by Trees, Figure 2) proved the existence of an inside-the-box tree consisting of $13$ connected line segments covering all the $27$ nodes of the ...
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