## Series extension – error message if nothing is to be evaluated

When I try to define this simple function

``````getCoeff(SeriesData(_, _, coeff_, _, _, _)) := coeff((2))
``````

I get an error:

``````SeriesData: Coefficient specification coeff_ in SeriesData(_,_,coeff_,_,_,_) is not a list.
``````

Can someone explain what's wrong with that? My understanding was that `SetDelayed` One should not complain about such things until I actually evaluate an expression that contains `getCoeff` it should not try to evaluate `Part(coeff,2)`,

No error is output for the similar function

``````f(g(_, _, x_, _, _, _)) := x((2))
``````

and also `getCoeff` works as expected:

``````getCoeff@Series(E^((Pi) x), {x, 0, 3})
``````

outputs $$pi$$,

I know the "right" way to manipulate `SeriesData`I just want to understand this case for its own sake.

## Algorithms – Create a priority search tree to determine the number of points in the range [-inf, qx] X [qy, qy’] from a series of points sorted by y-coordinates

A priority search tree can be created for a set of points P in time O (n log (n)). However, if the points are sorted by the y-coordinates, it takes O (n). I find algorithms for constructing the tree if the points are not sorted.

I found a way to do this as follows:

1. Construct a BST on the points.
Since the points are sorted, it will take O (n) time

2. Min-heap the BST on the x-coordinates
This will take theta (n) time

The total time complexity will be O (n).

Is this a valid approach to creating a priority search tree in O (n) time?

## cv.complex variables – In the Dirichlet series for \$ 1 / zeta (s) \$ for real \$ s \$ and the zeros of zeta

To the $$Re (s)> 1$$, it is well known that
$$frac {1} { zeta (s)} = sum_ {n = 1} ^ { infty} frac { mu (n)} {n ^ s}$$ Where $$mu$$ denotes the Mobius function and $$zeta$$ is the Riemann zeta function. I have heard that if the series on the right has an analytical continuation REAL $$s_ {0} in (1/2, 1)$$, then $$zeta (s) neq 0$$ for each $$s$$ With$$Re (s) = s_0$$, But since all non-trivial zeros of the zeta function are complex, why does the analytic continuation of the mentioned series occur? real Values ​​of $$s$$ something to do with that complex Zeros?

## Self-contained infinite series – Mathematics Stack Exchange

What do you call such an infinite sequence? $$(x_n) _ {n in Bbb N}: forall a = (x_i, x_ {i + 1}, x_ {i + 2}, …, x_ {i + j}) exists b = (x_k, x_ {k + 1}, x_ {k + 2}, …, x_ {k + j}); i not = k: a = b$$

For example, the 2-adic rating of 2n, https://oeis.org/A001511

In other words, every part of the sequence has a copy of itself in the sequence. In fact, each part would have an infinite number of copies of itself. I thought that was a fractal sequence, but that's not the case.

## Function series with Hermite polynomials

I wonder if there is a simpler form of the following series of functions, in which even Hermite polynomials are involved:
$$begin {equation} f (x) = e ^ {- frac {x ^ 2} {2}} cdot sum_ {n = 0} ^ { infty} , frac {1} {4 ^ nn!} cdot frac {H_ {2n} left (x right)} {a + n} end {equation}$$
Where $$a> 0$$,

The series seems to converge. It would be great to have a closed form for it. I expect a hump-shaped function with Cauchy-like fat tails.

## real analysis – uniform convergence of the \$ sum (nx) ^ {- 2} \$ series

I try to check the convergence of the series $$displaystyle sum_ {n = 1} ^ infty dfrac {1} {(nx) ^ 2}$$ on $$mathbb R – {0 }$$

I have tried the following:

If $$| x | 1$$ then $$(nx) ^ 2 geq n ^ 2$$ in order to $$dfrac {1} {(nx) ^ 2} leq dfrac {1} {n ^ 2}$$ and after the Weirstrass M-Test, the series is uniformly converging.

I am not sure how to proceed $$| x | <1$$, One thing I observe is $$f_n (1 / n) = 1$$ but I'm not sure if that helps.

## Form a box with a series of vectors

I have heard the term "box" of a series of vectors.
Here all elements of the vectors are a natural number or 0.

Could you give me a formal definition of a box?
This is,

A lot of vectors forms a box, if what?

## Probability – series of random variables with tendency to infinity P.a.s. and in L ^ 2

To let $$(X_n) n in mathbb {N}$$ be a sequence of random variables $$L ^ 2 ( Omega)$$ Review for everyone $$n$$ and for some positive real $$K$$ :
$$X_n geq 0 ; mathbb {P} -a.s.,$$
$$mathbb {E} (Xn) geq K> 0,$$
$$lim_ {n + infty} mathbb {E} (X_n ^ 2) = + infty$$
$$lim_ {n + infty} X_n = + infty ; mathbb {P} -. a.s.$$

Do we have :
$$lim_ {n + infty} mathbb {E} (X_n) = + infty.$$
If not, what would be a counterexample?

## Sequences and Series – Is there a deep philosophy or intuition behind the similarity between \$ pi / 4 \$ and \$ e ^ {- gamma} \$?

Here are a few examples of the similarity of Wikipedia, where the expressions differ only in characters.
I also came across other analogies.

{ begin {align} gamma & = int _ {0} ^ {1} int _ {0} ^ {1} { frac {x-1} {(1-xy) ln xy} } , dx , dy \ & = sum_ {n = 1} ^ { infty} left ({ frac {1} {n}} – ln { frac {n + 1} {n }} right). end {align}}

{ begin {align} ln { frac {4} { pi}} & = int _ {0} ^ {1} int _ {0} ^ {1} { frac {x-1 } {(1 + xy) ln xy}} , dx , dy \ & = sum_ {n = 1} ^ { infty} left ((- 1) ^ {n-1} left ({ frac {1} {n}} – ln { frac {n + 1} {n}} right) right). end {align}}

{ begin {align} gamma & = sum_ {n = 1} ^ { infty} { frac {N_ {1} (n) + N_ {0} (n)} {2n (2n + 1)}} \ ln { frac {4} { pi}} & = sum_ {n = 1} ^ { infty} { frac {N_ {1} (n) -N_ {0} (n)} {2n (2n + 1)}}, end {align}}

I wonder if there is an algebraic system there $$4e ^ {- gamma}$$ would play a similar role as what $$pi$$ plays, for example in complex numbers, or in a geometric system in which $$4e ^ {- gamma}$$ would play a special role, like $$pi$$ in Euclidean and Riemannian geometries.

## Calculus – Calculation of a series based on a parameter

From my textbook:

Calculation: $$sum_ {n = 1} ^ infty frac {10n ^ 2} {n ^ 3 + k ^ 3}$$

The problem is that I do not understand what they mean by "calculate". It looks like WolframAlpha does not have a nice closed formula. So you just want to show that the series diverges? If so, how can I do that?