Embedded Files
V** = 1÷0∞ = ∅⁻¹ = ÷⍬
⍴⍴⍴x = 1
Universal problem solver (of order): ? ⍨ x
*: There may be possibilities which never will occur (confer John 10:35).
**: The world is not enough: ∞∞ = (1÷0)∞ = 1∞÷0∞ = e÷∅ = e × V = eV.
***: My conjecture (John 10:30; 14:28).
(1/2)^n*Cos[Pi*13^n*x]; (* Weierstraß Alert! *)
Series[%, {n, Infinity, 0}] (* O[1/n]^2 = Series[O[n^(-1)]^2, {n, Infinity, 4}] *)
D[%, x]//Normal (* Magic *)
Integrate[%, x] === %%% (* LOL *)
dW[x_] = Sum[%%, {n, 1, Infinity}]; (* The differentiated Weierstrass function *)
dW[13/2]//N
ClearAll["Global`*"]
2^(O[1/n]^2 - n)*Cos[Pi*13^(n + O[1/n]^2)*x]
-Pi*(13/2)^n*Sin[Pi*13^n*x]
True
Sum::div: Sum does not converge.
NSum::emcon: Euler-Maclaurin sum failed to converge to requested error tolerance.
-1.17931079162605×10^13
Code to differentiate the Riemann function:
Sin[n^2*x]/n^2;
Series[%, {n, Infinity, 2}]//Normal
D[%, x]
dR[x_] = Sum[%, {n, 2, Infinity}]; (* The differentiated Riemann function *)
dR[2*Pi/9]//N
ClearAll["Global`*"]
Sin[n^2*x]/n^2
Cos[n^2*x]
Sum::div: Sum does not converge.
NIntegrate::deorela: The relative error 2.000419701099671` is larger than expected for the integrand Cos[2*Pi/9]*Cos[2*n*Pi/9]/2/Sqrt[n + 289] over {0,Infinity} with DoubleExponentialOscillatory method and automatic tuning parameters, TuningParameters -> {10,5}. The integration will proceed with TuningParameters -> {1,5}.
NIntegrate::deodiv: DoubleExponentialOscillatory returns a finite integral estimate, but the integral might be divergent.
NIntegrate::deodiv: DoubleExponentialOscillatory returns a finite integral estimate, but the integral might be divergent.
NSum::emcon: Euler-Maclaurin sum failed to converge to requested error tolerance.
5.095393132022091
m = 3;
n0 = 1;
n^m; (* The little divergent calculator (the standard part; st) :) *)
If[% =!= Sst[%], Sst[%], %];
Sum[%, {n, n0, k}]
Limit[%, k -> Infinity];
If[NumericQ[%], %, Integrate[%%, {k, -1, 0}]]
Zeta[-m]
ClearAll["Global`*"]
(k + 1)^2*k^2/4
1/120
1/120
n0 = 1;
1/n; (* The Sum All Series (SAS) *)
If[% =!= Sst[%], Sst[%], %];
Off[Series::serlim]
Series[%%, {n, Infinity, n}, Analytic -> False]//Normal;
On[Series::serlim]
Sum[%%, {n, n0, Infinity}]//N; (* “//Quiet” also works *)
If[NumericQ[%], %, Series[Out[-5], {n, Infinity, n0}, Analytic -> False]//Normal//FullSimplify//ToRadicals]
ClearAll["Global`*"]
Sum::div: Sum does not converge.
NIntegrate::slwcon: Numerical integration converging too slowly; suspect one of the following: singularity, value of the integration is 0, highly oscillatory integrand, or WorkingPrecision too small.
NIntegrate::ncvb: NIntegrate failed to converge to prescribed accuracy after 9 recursive bisections in n near {n} = {8.169069784662288×10^224}. NIntegrate obtained 191609.51762979227` and 160378.5178102709` for the integral and error estimates.
191612.8674343063
An extended number is in the form 𝜏 = x∞ⁿ + y where the infinite part ∞ⁿ (0 < n < ∞) is a number with the property
(±∞)ⁿ × 0ⁿ = (±1)ⁿ (1), and a result of the indivisibility of infinite elements (2) is ƒ(x) = x∞ ⇒ ƒ|ℚ : ℚ → ℤ∞; the sign is located in the dividend*.
The theory is noncommutative to keep compatibility with multiplication by zero: x∞ⁿ0ⁿ = x(∞0)ⁿ = x, x ∉ ℚ and is only true with the retention of finite elements; i.e. with multiplication from the right if both sides are raised to the power of 0 as only then Euler’s identity holds, this also solves the problem
“which y is the solution to y × 0 = x?”: y = x∞, x ∉ ℚ. It is also nondistributive*:
∞ × (0 ± 0) = 1 ≠ ∞ × 0 + ∞ × 0 = 2 ≠ ∞ × 0 − ∞ × 0 = 0; zeros are cancelled out inside parentheses separately. A solution to x ∉ ℚ is that fractions as constants can be expressed in another way with the infinite part: ⁿ√∞.
“The Seven Spirits of God”: ∞ − ∞, 0 × ∞, 0 ÷ 0, ∞ ÷ ∞, 0⁰, ∞⁰ and 1∞ which equal 1, 1, 1, 1, 1, 1, and e (3). Fallacies are corrected by first multiplying by 0 if also dividing by 0*: 0 × a ÷ 0 = 0 × b ÷ 0 ⇒ (0 × a) ÷ 0 = (0 × b) ÷ 0 = 0 ÷ 0 = 1.
Factorials: n! = n × (n − 1)! ⇒ 0! = 0 × (−1)! = 0 × ∞ = 1, −(n=1)∑1 ÷ n = ln (0) ≠ −ln (∞) − γ, ε = 1÷ω ⇔ ω = 1÷ε.
*: In this case.
{c, n, k, b} = {10, n, n, n}; (* {n, k, b} = n *)
Unprotect[Divide];
Divide[0, 0] = 1;
f = n*b^(Floor[Log[b, k]] + 1) + k//FullSimplify//ToRadicals (* {n, k} → nk *)
Divide[0, 0] = .;
Protect[Divide];
s = Sum[f, {n, 1, m}]//FullSimplify//ToRadicals
ParallelTable[f, {n, -c, c}, Method -> "FinestGrained"]
ParallelTable[s, {m, -c, c}, Method -> "FinestGrained"]
ClearAll["Global`*"]
n^3 + n
(m^2 + m + 2)*(m + 1)*m/4
{-1010, -738, -520, -350, -222, -130, -68, -30, -10, -2, 0, 2, 10, 30, 68, 130, 222, 350, 520, 738, 1010}
{2070, 1332, 812, 462, 240, 110, 42, 12, 2, 0, 0, 2, 12, 42, 110, 240, 462, 812, 1332, 2070, 3080}
Settings for Mathematica:
Settings for Mathematica:
SetSystemOptions["BooleanComputationOptions" -> "BDDReordering" -> False];
SetSystemOptions["BooleanComputationOptions" -> "MinimizationManualReordering" -> True];
SetSystemOptions["BooleanComputationOptions" -> "OrderingFunction" -> None];
SetSystemOptions["BooleanComputationOptions" -> "ZDDReordering" -> False];
Unprotect[$PerformanceGoal];
Unprotect[$RecursionLimit];
$PerformanceGoal = "Quality";
$RecursionLimit = Infinity;
Protect[$RecursionLimit];
Protect[$PerformanceGoal];
Needs["Developer`"]
Needs["ComputerArithmetic`"]
Needs["Quaternions`"] (* Quaternion[R, I, J, K], Mod[n + 2, -2] + Mod[n + 1, 2] = (-1)^n *)
PlusMinus[x_] = {x, -x};
PlusMinus[x_, y_] = {x + y, x - y};
MinusPlus[x_] = {-x, x};
MinusPlus[x_, y_] = {x - y, x + y};
AppendTo[$ContextPath, "special`"];
special`Ii[x_] = Ceiling[x] - Floor[x]; (* A Gaussian integer indicator (Ii, 0: Gaussian integer) *)
special`Sst[f_] := Refine[f, Element[x | y | z, Complexes] && Element[n | k | m, NonNegativeIntegers]] (* My series sequence transformer (Sst) *)
SetAttributes[{PlusMinus, MinusPlus, special`Ii, special`Sst}, {Listable, NumericFunction, Protected}]
f[x_] := f[x] = x (* Conditional assignment *)
Update[]
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