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 SeriesDataI 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.

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.