digerdiga

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11 years, 81 days

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These are questions asked by digerdiga

The following product gives 0

restart;

product(1-1/k, k = 2 .. infinity);


                               0

However when I expand the product

1 - 1/2 - 1/3 - 1/4 - ... + 1/2*1/3 + 1/2*1/4 + ... + 1/3*1/4 + ... + triple products + quadruple products + and so forth...

Now the double, triple, quadruple, and so forth sums of products converge.

The 1/2 + 1/3 + 1/4 + ... nevertheless diverges, so why does maple give me 0?

Suppose I have a subset of R^2, say (x,y) € [0,1] x [0,1]

Now I map

x=r*exp(t)

y=r*exp(-t)

Is it possible to do a 2d contourplot to just see where [0,1]x[0,1] is mapped onto ?

I tried to integrate

int((1-x^floor(u))/((1-x)*u^2), u = 1 .. infinity, numeric)

where x=-1. The result should be log 2 = 0.6931471806. However it gives me 0.6687714032.

When using a numeric cut off, the result improves, so what is the issue here?

6687714032

In Physics package there is this compact notation X=(t,x,y,z)

Is there something similar in the VectorCalculus packge?

For example

restart;

with(VectorCalculus);

SetCoordinates(cartesian[x, y, z]);

v := VectorField([vx, vy, vz]);

Jacobian(v);

 

I don't explicitly want to write the arguments (x,y,z) of the functions vx,vy,vz everytime.

So I have this system of equations with which I am not sure if the result is the same or not using "series" and "limit" or what is going on here.

I hope it is clear what I mean.


 

restart; with(MathematicalFunctions); Assume(k__2H2O > 0, `k__HA+OH` > 0, `k__A+H2O` > 0, `k__H3O+OH` > 0, `k__HA+H2O` > 0, `k__H3O+A` > 0, HA__0 > 0, H2O > 0); sys := k__2H2O*H2O^2+`k__A+H2O`*H2O*(HA__0-HA)-(H3O*`k__H3O+OH`+HA*`k__HA+OH`)*OH = 0, k__2H2O*H2O^2+`k__HA+H2O`*H2O*HA-(`k__H3O+A`*(HA__0-HA)+`k__H3O+OH`*OH)*H3O = 0, (H2O*`k__HA+H2O`+OH*`k__HA+OH`)*HA-(H2O*`k__A+H2O`+H3O*`k__H3O+A`)*(HA__0-HA) = 0; sys := `~`[simplify]([eval(eval(sys, HA = HA__0+OH-H3O), HA__0 = x__HA0*H2O)]); sol := solve(sys, [OH, H3O]); sol := sol[1]; OH__sol := simplify(rhs(sol[1])); H3O__sol := simplify(rhs(sol[2])); simplify(OH__sol*H3O__sol); OHH3O := simplify(limit(%, `k__HA+OH` = 0)); series(OHH3O, x__HA0 = 0, 2); collect(convert(%, polynom), x__HA0, simplify, factor); r1 := limit(%, x__HA0 = 0); r2 := radnormal(limit(OHH3O, x__HA0 = 0)); simplify(r1-r2)

[`&Intersect`, `&Minus`, `&Union`, Assume, Coulditbe, Evalf, Get, Is, SearchFunction, Sequences, Series]

 

{H2O::(RealRange(Open(0), infinity))}, {HA__0::(RealRange(Open(0), infinity))}, {k__2H2O::(RealRange(Open(0), infinity))}, {`k__A+H2O`::(RealRange(Open(0), infinity))}, {`k__H3O+A`::(RealRange(Open(0), infinity))}, {`k__H3O+OH`::(RealRange(Open(0), infinity))}, {`k__HA+H2O`::(RealRange(Open(0), infinity))}, {`k__HA+OH`::(RealRange(Open(0), infinity))}

 

k__2H2O*H2O^2+`k__A+H2O`*H2O*(HA__0-HA)-(H3O*`k__H3O+OH`+HA*`k__HA+OH`)*OH = 0, k__2H2O*H2O^2+`k__HA+H2O`*H2O*HA-(`k__H3O+A`*(HA__0-HA)+`k__H3O+OH`*OH)*H3O = 0, (H2O*`k__HA+H2O`+OH*`k__HA+OH`)*HA-(H2O*`k__A+H2O`+H3O*`k__H3O+A`)*(HA__0-HA) = 0

 

[-OH^2*`k__HA+OH`+((-x__HA0*`k__HA+OH`-`k__A+H2O`)*H2O+H3O*(`k__HA+OH`-`k__H3O+OH`))*OH+k__2H2O*H2O^2+`k__A+H2O`*H2O*H3O = 0, (x__HA0*`k__HA+H2O`+k__2H2O)*H2O^2+`k__HA+H2O`*(OH-H3O)*H2O+(-`k__H3O+A`*H3O+OH*(`k__H3O+A`-`k__H3O+OH`))*H3O = 0, H2O^2*x__HA0*`k__HA+H2O`+((x__HA0*`k__HA+OH`+`k__A+H2O`+`k__HA+H2O`)*OH-H3O*(`k__A+H2O`+`k__HA+H2O`))*H2O+(OH-H3O)*(H3O*`k__H3O+A`+OH*`k__HA+OH`) = 0]

 

-RootOf(-x__HA0*`k__A+H2O`^2*`k__HA+H2O`+k__2H2O^2*`k__H3O+A`-k__2H2O*`k__A+H2O`^2-k__2H2O*`k__A+H2O`*`k__HA+H2O`+(2*x__HA0*`k__A+H2O`*`k__H3O+OH`*`k__HA+H2O`-k__2H2O*`k__A+H2O`*`k__H3O+A`+k__2H2O*`k__A+H2O`*`k__H3O+OH`+k__2H2O*`k__H3O+OH`*`k__HA+H2O`)*_Z+(-x__HA0*`k__H3O+OH`^2*`k__HA+H2O`-k__2H2O*`k__H3O+A`*`k__H3O+OH`+`k__A+H2O`^2*`k__H3O+OH`+`k__A+H2O`*`k__H3O+OH`*`k__HA+H2O`)*_Z^2+(`k__A+H2O`*`k__H3O+A`*`k__H3O+OH`-`k__A+H2O`*`k__H3O+OH`^2-`k__H3O+OH`^2*`k__HA+H2O`)*_Z^3)*H2O^2*(-`k__A+H2O`*RootOf(-x__HA0*`k__A+H2O`^2*`k__HA+H2O`+k__2H2O^2*`k__H3O+A`-k__2H2O*`k__A+H2O`^2-k__2H2O*`k__A+H2O`*`k__HA+H2O`+(2*x__HA0*`k__A+H2O`*`k__H3O+OH`*`k__HA+H2O`-k__2H2O*`k__A+H2O`*`k__H3O+A`+k__2H2O*`k__A+H2O`*`k__H3O+OH`+k__2H2O*`k__H3O+OH`*`k__HA+H2O`)*_Z+(-x__HA0*`k__H3O+OH`^2*`k__HA+H2O`-k__2H2O*`k__H3O+A`*`k__H3O+OH`+`k__A+H2O`^2*`k__H3O+OH`+`k__A+H2O`*`k__H3O+OH`*`k__HA+H2O`)*_Z^2+(`k__A+H2O`*`k__H3O+A`*`k__H3O+OH`-`k__A+H2O`*`k__H3O+OH`^2-`k__H3O+OH`^2*`k__HA+H2O`)*_Z^3)+k__2H2O)/(-`k__H3O+OH`*RootOf(-x__HA0*`k__A+H2O`^2*`k__HA+H2O`+k__2H2O^2*`k__H3O+A`-k__2H2O*`k__A+H2O`^2-k__2H2O*`k__A+H2O`*`k__HA+H2O`+(2*x__HA0*`k__A+H2O`*`k__H3O+OH`*`k__HA+H2O`-k__2H2O*`k__A+H2O`*`k__H3O+A`+k__2H2O*`k__A+H2O`*`k__H3O+OH`+k__2H2O*`k__H3O+OH`*`k__HA+H2O`)*_Z+(-x__HA0*`k__H3O+OH`^2*`k__HA+H2O`-k__2H2O*`k__H3O+A`*`k__H3O+OH`+`k__A+H2O`^2*`k__H3O+OH`+`k__A+H2O`*`k__H3O+OH`*`k__HA+H2O`)*_Z^2+(`k__A+H2O`*`k__H3O+A`*`k__H3O+OH`-`k__A+H2O`*`k__H3O+OH`^2-`k__H3O+OH`^2*`k__HA+H2O`)*_Z^3)+`k__A+H2O`)

 

-(k__2H2O*`k__H3O+A`^2-2*`k__A+H2O`^2*`k__H3O+A`+`k__A+H2O`^2*`k__H3O+OH`-2*`k__A+H2O`*`k__H3O+A`*`k__HA+H2O`+2*`k__A+H2O`*`k__H3O+OH`*`k__HA+H2O`+`k__H3O+OH`*`k__HA+H2O`^2)*`k__A+H2O`*`k__HA+H2O`*H2O^2*x__HA0/((`k__A+H2O`*`k__H3O+A`-`k__A+H2O`*`k__H3O+OH`-`k__H3O+OH`*`k__HA+H2O`)*(k__2H2O*`k__H3O+A`^2-`k__A+H2O`^2*`k__H3O+OH`-2*`k__A+H2O`*`k__H3O+OH`*`k__HA+H2O`-`k__H3O+OH`*`k__HA+H2O`^2))-(k__2H2O*`k__H3O+A`-`k__A+H2O`^2-`k__A+H2O`*`k__HA+H2O`)*H2O^2*(`k__A+H2O`+`k__HA+H2O`)/(`k__H3O+A`*(`k__A+H2O`*`k__H3O+A`-`k__A+H2O`*`k__H3O+OH`-`k__H3O+OH`*`k__HA+H2O`))

 

-(k__2H2O*`k__H3O+A`-`k__A+H2O`^2-`k__A+H2O`*`k__HA+H2O`)*H2O^2*(`k__A+H2O`+`k__HA+H2O`)/(`k__H3O+A`*(`k__A+H2O`*`k__H3O+A`-`k__A+H2O`*`k__H3O+OH`-`k__H3O+OH`*`k__HA+H2O`))

 

k__2H2O*H2O^2/`k__H3O+OH`

 

-`k__A+H2O`*(-(`k__A+H2O`+`k__HA+H2O`)^2*`k__H3O+OH`+k__2H2O*`k__H3O+A`^2)*H2O^2/(`k__H3O+OH`*`k__H3O+A`*((-`k__A+H2O`-`k__HA+H2O`)*`k__H3O+OH`+`k__A+H2O`*`k__H3O+A`))

(1)

``


 

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