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These are answers submitted by Mariusz Iwaniuk

A test example:...

Mariusz Iwaniuk 1571
May 19 2018
1 4

Maybe this helps:

0.https://mapleprimes.com/questions/223689-Comparing-RK4-And-RK45-How-To-Solve

1. https://www.mapleprimes.com/questions/223703-RungeKutta-4th-Order

2.https://www.mapleprimes.com/questions/221882-Maple-Code-For-4th-Order-Rungekutta

3.https://www.mapleprimes.com/questions/220556-How-Do-I-Implement-RungeKutta-Of-Order.

A test example:


 

> 

restart
> 

dsys := {diff(x(t), t) = y(t), diff(y(t), t) = x(t)+y(t), x(0) = 2, y(0) = 1}

{diff(x(t), t) = y(t), diff(y(t), t) = x(t)+y(t), x(0) = 2, y(0) = 1}

 (1)
> 

sol := dsolve(dsys, numeric, method = classical[rk4], stepsize = .1)

proc (x_classical) local _res, _dat, _vars, _solnproc, _xout, _ndsol, _pars, _n, _i; option `Copyright (c) 2000 by Waterloo Maple Inc. All rights reserved.`; if 1 < nargs then error "invalid input: too many arguments" end if; _EnvDSNumericSaveDigits := Digits; Digits := 15; if _EnvInFsolve = true then _xout := evalf[_EnvDSNumericSaveDigits](x_classical) else _xout := evalf(x_classical) end if; _dat := Array(1..4, {(1) = proc (_xin) local _xout, _n, _y0, _yn, _yl, _ctl, _octl, _reinit, _errcd, _fcn, _i, _yini, _pars, _ini, _par; option `Copyright (c) 2002 by the University of Waterloo. All rights reserved.`; table( [( "complex" ) = false ] ) _xout := _xin; _ctl := Array(1..9, {(1) = 2, (2) = 0., (3) = 0., (4) = .1, (5) = 50000, (6) = 0, (7) = 4, (8) = 1, (9) = 0}); _octl := Array(1..9, {(1) = 2, (2) = 0., (3) = 0., (4) = .1, (5) = 50000, (6) = 0, (7) = 4, (8) = 1, (9) = 0}); _n := trunc(_ctl[1]); _yini := Array(0..2, {(1) = 0., (2) = 2.}); _y0 := Array(0..2, {(1) = 0., (2) = 2.}); _yl := Array(1..2, {(1) = 0, (2) = 0}); _yn := Array(1..2, {(1) = 0, (2) = 0}); _fcn := proc (N, X, Y, YP) option `[Y[1] = x(t), Y[2] = y(t)]`; YP[2] := Y[1]+Y[2]; YP[1] := Y[2]; 0 end proc; _pars := []; if not type(_xout, 'numeric') then if member(_xout, ["start", "left", "right"]) then return _y0[0] elif _xout = "method" then return "classical" elif _xout = "numfun" then return round(_ctl[6]) elif _xout = "initial" then return [seq(_yini[_i], _i = 0 .. _n)] elif _xout = "parameters" then return [seq(_yini[_n+_i], _i = 1 .. nops(_pars))] elif _xout = "initial_and_parameters" then return [seq(_yini[_i], _i = 0 .. _n)], [seq(_yini[_n+_i], _i = 1 .. nops(_pars))] elif _xout = "last" then if _ctl[3]-_y0[0] <> 0. then _xout := _ctl[3] elif _ctl[2]-_y0[0] <> 0. then _xout := _ctl[2] else error "no information is available on last computed point" end if elif _xout = "enginedata" then return eval(_octl, 1) elif _xout = "function" then return eval(_fcn, 1) elif type(_xin, `=`) and type(rhs(_xin), 'list') and member(lhs(_xin), {"initial", "parameters", "initial_and_parameters"}) then _ini, _par := [], []; if lhs(_xin) = "initial" then _ini := rhs(_xin) elif lhs(_xin) = "parameters" then _par := rhs(_xin) elif select(type, rhs(_xin), `=`) <> [] then _par, _ini := selectremove(type, rhs(_xin), `=`) elif nops(rhs(_xin)) < nops(_pars)+1 then error "insufficient data for specification of initial and parameters" else _par := rhs(_xin)[-nops(_pars) .. -1]; _ini := rhs(_xin)[1 .. -nops(_pars)-1] end if; _xout := lhs(_xout); if _par <> [] then `dsolve/numeric/process_parameters`(_n, _pars, _par, _yini) end if; if _ini <> [] then `dsolve/numeric/process_initial`(_n, _ini, _yini, _pars) end if; if _pars <> [] then _par := {seq(rhs(_pars[_i]) = _yini[_n+_i], _i = 1 .. nops(_pars))}; for _i from 0 to _n do _y0[_i] := subs(_par, _yini[_i]) end do; for _i from _n+1 to _n+nops(_pars) do _y0[_i] := _yini[_i] end do; for _i to nops(_pars) do _yn[_n+_i] := _y0[_n+_i] end do else for _i from 0 to _n do _y0[_i] := _yini[_i] end do end if; _octl[2] := _y0[0]; _octl[3] := _y0[0]; for _i to 9 do _ctl[_i] := _octl[_i] end do; for _i to _n+nops(_pars) do _yl[_i] := _y0[_i] end do; if _Env_smart_dsolve_numeric = true and type(_y0[0], 'numeric') then procname("right") := _y0[0]; procname("left") := _y0[0] end if; if _xout = "initial" then return [seq(_yini[_i], _i = 0 .. _n)] elif _xout = "parameters" then return [seq(_yini[_n+_i], _i = 1 .. nops(_pars))] else return [seq(_yini[_i], _i = 0 .. _n), op(_pars)], [seq(_yini[_n+_i], _i = 1 .. nops(_pars))] end if else return "procname" end if end if; if _xout-_y0[0] = 0. then return [seq(_y0[_i], _i = 0 .. _n)] end if; _reinit := false; if _xin <> "last" then if 0 < 0 and `dsolve/numeric/checkglobals`(0, table( [ ] ), _pars, _n, _yini) then _reinit := true; if _pars <> [] then _par := {seq(rhs(_pars[_i]) = _yini[_n+_i], _i = 1 .. nops(_pars))}; for _i from 0 to _n do _y0[_i] := subs(_par, _yini[_i]) end do; for _i from _n+1 to _n+nops(_pars) do _y0[_i] := _yini[_i] end do; for _i to nops(_pars) do _yn[_n+_i] := _y0[_n+_i] end do else for _i from 0 to _n do _y0[_i] := _yini[_i] end do end if end if; if _pars <> [] and select(type, {seq(_yini[_n+_i], _i = 1 .. nops(_pars))}, 'undefined') <> {} then error "parameters must be initialized before solution can be computed" end if end if; if not _reinit and _xout-_ctl[3] = 0. then [_ctl[3], seq(_yn[_i], _i = 1 .. _n)] elif not _reinit and _xout-_ctl[2] = 0. then [_ctl[2], seq(_yl[_i], _i = 1 .. _n)] else if _ctl[2]-_y0[0] = 0. or sign(_ctl[2]-_y0[0]) <> sign(_xout-_ctl[2]) or abs(_xout-_y0[0]) < abs(_xout-_ctl[2]) or _reinit then for _i to 9 do _ctl[_i] := _octl[_i] end do; for _i to _n+nops(_pars) do _yl[_i] := _y0[_i] end do; for _i to nops(_pars) do _yn[_n+_i] := _y0[_n+_i] end do end if; _ctl[3] := _xout; if Digits <= evalhf(Digits) then try _errcd := evalhf(`dsolve/numeric/classical`(_fcn, var(_ctl), _y0[0], var(_yl), var(_yn), Array(1..2, {(1) = 0, (2) = 0}), Array(1..2, {(1) = 0, (2) = 0}), Array(1..2, {(1) = 0, (2) = 0}), Array(1..2, {(1) = 0, (2) = 0}), var(Array(1..3, 1..2, {(1, 1) = 0, (1, 2) = 0, (2, 1) = 0, (2, 2) = 0, (3, 1) = 0, (3, 2) = 0})))) catch: userinfo(2, `dsolve/debug`, print(`Exception in classical:`, [lastexception])); if searchtext('evalhf', lastexception[2]) <> 0 or searchtext('real', lastexception[2]) <> 0 or searchtext('hardware', lastexception[2]) <> 0 then _errcd := `dsolve/numeric/classical`(_fcn, _ctl, _y0[0], _yl, _yn, Array(1..2, {(1) = 0, (2) = 0}), Array(1..2, {(1) = 0, (2) = 0}), Array(1..2, {(1) = 0, (2) = 0}), Array(1..2, {(1) = 0, (2) = 0}), Array(1..3, 1..2, {(1, 1) = 0, (1, 2) = 0, (2, 1) = 0, (2, 2) = 0, (3, 1) = 0, (3, 2) = 0})) else _ctl[3] := _y0[0]; _errcd := [lastexception][2 .. -1] end if end try else try _errcd := `dsolve/numeric/classical`(_fcn, _ctl, _y0[0], _yl, _yn, Array(1..2, {(1) = 0, (2) = 0}), Array(1..2, {(1) = 0, (2) = 0}), Array(1..2, {(1) = 0, (2) = 0}), Array(1..2, {(1) = 0, (2) = 0}), Array(1..3, 1..2, {(1, 1) = 0, (1, 2) = 0, (2, 1) = 0, (2, 2) = 0, (3, 1) = 0, (3, 2) = 0})) catch: _ctl[3] := _y0[0]; _errcd := [lastexception][2 .. -1] end try end if; if type(_errcd, 'list') or _errcd < 0 then userinfo(2, {dsolve, `dsolve/classical`}, `Last values returned:`); userinfo(2, {dsolve, `dsolve/classical`}, ` t =`, _ctl[2]); userinfo(2, {dsolve, `dsolve/classical`}, ` y =`, _yl[1]); for _i from 2 to _n do userinfo(2, {dsolve, `dsolve/classical`}, ` `, _yl[_i]) end do; if type(_errcd, 'list') then error op(_errcd) elif _errcd+1. = 0. then Rounding := `if`(_y0[0] < _xout, -infinity, infinity); error "cannot evaluate the solution past %1, maxfun limit exceeded (see ?dsolve,maxfun for details)", evalf[8](_ctl[3]) else Rounding := `if`(_y0[0] < _xout, -infinity, infinity); error "cannot evaluate the solution past %1, unknown error code returned from classical %2", evalf[8](_ctl[3]), trunc(_errcd) end if end if; if _Env_smart_dsolve_numeric = true then if _y0[0] < _xout and procname("right") < _xout then procname("right") := _xout elif _xout < _y0[0] and _xout < procname("left") then procname("left") := _xout end if end if; [_xout, seq(_yn[_i], _i = 1 .. _n)] end if end proc, (2) = Array(0..0, {}), (3) = [t, x(t), y(t)], (4) = []}); _vars := _dat[3]; _pars := map(rhs, _dat[4]); _n := nops(_vars)-1; _solnproc := _dat[1]; if not type(_xout, 'numeric') then if member(x_classical, ["start", 'start', "method", 'method', "left", 'left', "right", 'right', "leftdata", "rightdata", "enginedata", "eventstop", 'eventstop', "eventclear", 'eventclear', "eventstatus", 'eventstatus', "eventcount", 'eventcount', "laxtol", 'laxtol', "numfun", 'numfun', NULL]) then _res := _solnproc(convert(x_classical, 'string')); if 1 < nops([_res]) then return _res elif type(_res, 'array') then return eval(_res, 1) elif _res <> "procname" then return _res end if elif member(x_classical, ["last", 'last', "initial", 'initial', "parameters", 'parameters', "initial_and_parameters", 'initial_and_parameters', NULL]) then _xout := convert(x_classical, 'string'); _res := _solnproc(_xout); if _xout = "parameters" then return [seq(_pars[_i] = _res[_i], _i = 1 .. nops(_pars))] elif _xout = "initial_and_parameters" then return [seq(_vars[_i+1] = [_res][1][_i+1], _i = 0 .. _n), seq(_pars[_i] = [_res][2][_i], _i = 1 .. nops(_pars))] else return [seq(_vars[_i+1] = _res[_i+1], _i = 0 .. _n)] end if elif type(_xout, `=`) and member(lhs(_xout), ["initial", 'initial', "parameters", 'parameters', "initial_and_parameters", 'initial_and_parameters', NULL]) then _xout := convert(lhs(x_classical), 'string') = rhs(x_classical); if type(rhs(_xout), 'list') then _res := _solnproc(_xout) else error "initial and/or parameter values must be specified in a list" end if; if lhs(_xout) = "initial" then return [seq(_vars[_i+1] = _res[_i+1], _i = 0 .. _n)] elif lhs(_xout) = "parameters" then return [seq(_pars[_i] = _res[_i], _i = 1 .. nops(_pars))] else return [seq(_vars[_i+1] = [_res][1][_i+1], _i = 0 .. _n), seq(_pars[_i] = [_res][2][_i], _i = 1 .. nops(_pars))] end if elif type(_xout, `=`) and member(lhs(_xout), ["eventdisable", 'eventdisable', "eventenable", 'eventenable', "eventfired", 'eventfired', "direction", 'direction', NULL]) then return _solnproc(convert(lhs(x_classical), 'string') = rhs(x_classical)) elif _xout = "solnprocedure" then return eval(_solnproc) elif _xout = "sysvars" then return _vars end if; if procname <> unknown then return ('procname')(x_classical) else _ndsol := 1; _ndsol := _ndsol; _ndsol := pointto(_dat[2][0]); return ('_ndsol')(x_classical) end if end if; try _res := _solnproc(_xout); [seq(_vars[_i+1] = _res[_i+1], _i = 0 .. _n)] catch: error end try end proc

 (2)
> 

plots:-odeplot(sol, [[t, x(t)], [t, y(t)]], t = 0 .. 2, color = ["Red", "Blue"])

 
> 

NULL


 

Download testODE.mw

 

Is build in....

Mariusz Iwaniuk 1571
May 19 2018
0 0

This package is build-in Maple,you don't need to download anything:

Execute:

with(Student[Basics]);

LinearSolveSteps((x+1)/(2*y*z) = 4*y^2/z+3*x/y, x);

LinearSolveSteps function have some limitation only  for linear equations give steps.

LinearSolveSteps("x^2 -2*x+1 = 0", x); #Nonlinear equation. It doesn't work.

 

 

No solution....

Mariusz Iwaniuk 1571
May 17 2018
0 0

Eliminating variable m1 and m2 from equations, then we have only eq. with 2 variables k1 and k2.

Ploting we see they do not intersect anywhere, even at point {0,0}.


 

> 

restart
> 

sys := {(k1*m1+k1*m2+k2*m1+sqrt(k1^2*m1^2+2*k1^2*m1*m2+k1^2*m2^2+2*k1*k2*m1^2-2*k1*k2*m1*m2+k2^2*m1^2))/(2*m1*m2) = (2*Pi*8.78)^2, .8347192842*k1/m1 = (2*Pi*4.8515)^2, -(-30.85287127*k1-k2)/m2 = (2*Pi*8.78)^2, -(-k1*m1-k1*m2-k2*m1+sqrt(k1^2*m1^2+2*k1^2*m1*m2+k1^2*m2^2+2*k1*k2*m1^2-2*k1*k2*m1*m2+k2^2*m1^2))/(2*m1*m2) = (2*Pi*4.8515)^2}

{.8347192842*k1/m1 = 929.2055780, -(-30.85287127*k1-k2)/m2 = 3043.328048, -(1/2)*(-k1*m1-k1*m2-k2*m1+(k1^2*m1^2+2*k1^2*m1*m2+k1^2*m2^2+2*k1*k2*m1^2-2*k1*k2*m1*m2+k2^2*m1^2)^(1/2))/(m1*m2) = 929.2055780, (1/2)*(k1*m1+k1*m2+k2*m1+(k1^2*m1^2+2*k1^2*m1*m2+k1^2*m2^2+2*k1*k2*m1^2-2*k1*k2*m1*m2+k2^2*m1^2)^(1/2))/(m1*m2) = 3043.328048}

 (1)
> 

eq2 := eliminate(sys, {m1, m2})

[{m1 = 0.8983149735e-3*k1, m2 = 0.1013787235e-1*k1+0.3285876462e-3*k2}, {-0.2219750257e-1*k1^2-0.2848636636e-3*k1*k2+(1/2)*(0.1217974307e-3*k1^4-0.9347355846e-5*k1^3*k2+0.3245892274e-6*k1^2*k2^2)^(1/2), 0.2944183889e-2*k1^2-0.3391728651e-3*k1*k2+(1/2)*(0.1217974307e-3*k1^4-0.9347355846e-5*k1^3*k2+0.3245892274e-6*k1^2*k2^2)^(1/2)}]

 (2)
> 

plots:-implicitplot([eq2[2, 1] = 0, eq2[2, 2] = 0], k1 = -1 .. 1, k2 = -10 .. 10, view = [-1 .. 1, -10 .. 10], gridrefine = 5, color = ["Red", "Blue"], rational, thickness = 3)

 
> 

plots:-implicitplot([eq2[2, 1] = 0, eq2[2, 2] = 0], k1 = -0.1e-3 .. 0.1e-3, k2 = -0.1e-1 .. 0.1e-1, view = [-0.1e-3 .. 0.1e-3, -0.1e-1 .. 0.1e-1], gridrefine = 5, color = ["Red", "Blue"], rational, thickness = 3)

 
> 

``


 

Download No_solutions.mw

....

Mariusz Iwaniuk 1571
May 13 2018
0 0 
You must add 4 more (intial,boundary) condition if you what to solve by numeric method.
e.g. :{ U(0, t) = 1, U(1, t) = 0, V(0, t) = 0, V(1, t) = 0}


 

> 

PDESYS := [diff(U(x, t), t)-(diff(U(x, t), x, x))-2*U(x, t)*(diff(U(x, t), x))+diff(U(x, t)*V(x, t), x), diff(V(x, t), t)-(diff(V(x, t), x, x))-2*V(x, t)*(diff(V(x, t), x))+diff(U(x, t)*V(x, t), x)]

[diff(U(x, t), t)-(diff(diff(U(x, t), x), x))-2*U(x, t)*(diff(U(x, t), x))+(diff(U(x, t), x))*V(x, t)+U(x, t)*(diff(V(x, t), x)), diff(V(x, t), t)-(diff(diff(V(x, t), x), x))-2*V(x, t)*(diff(V(x, t), x))+(diff(U(x, t), x))*V(x, t)+U(x, t)*(diff(V(x, t), x))]

 (1)
> 

NULL
> 

IBC := [U(x, 0) = sin(x), V(x, 0) = sin(x), U(0, t) = 1, U(1, t) = 0, V(0, t) = 0, V(1, t) = 0]

[U(x, 0) = sin(x), V(x, 0) = sin(x), U(0, t) = 1, U(1, t) = 0, V(0, t) = 0, V(1, t) = 0]

 (2)
> 

sol := pdsolve(PDESYS, IBC, numeric)

_m489011961280

 (3)
> 

p1 := sol:-plot(V(x, t), t = 1/8, numpoints = 100, x = 0 .. 1, color = ["Blue"], legend = ["V(x,t)"])
> 

p2 := sol:-plot(U(x, t), t = 1/20, numpoints = 100, x = 0 .. 1, color = ["Green"], legend = ["U(x,t)"])
> 

plots:-display({p1, p2})

 
> 

sol:-plot3d(U(x, t), t = 0 .. 1, x = 0 .. 1, grid = [100, 100])

 
> 

sol:-plot3d(V(x, t), t = 0 .. 1, x = 0 .. 1, grid = [100, 100])

 
> 

``


Executed in Maple 2018 !!!

Download pdesys.mw

Procedure with IF...

Mariusz Iwaniuk 1571
May 12 2018
1 0 
Y := proc (delta, b, n) 
if n = 0 then 1/(b^2+1) 
elif n = 1 then arctan(1/b) 
elif n = 2 then 1+(1/2)*ln(b^2+1)-b*arctan(1/b) 
elif n = 3 then delta-(3/2)*b-(1/2)*b*ln(b^2+1)+(1/2)*arctan(1/b)*(b^2-1) 
elif n = 4 then (1/2)*delta^2-b*delta+(11/12)*b^2-11/36+(1/12)*ln(b^2+1)*(3*b^2-1)+(1/6)*b*arctan(1/b)*(3-b^2) 
elif n = 5 then (1/3)*delta^3-(1/2)*delta^2*b+(1/6)*delta*(3*b^2-1)+(25/72)*b-(25/72)*b^3+(1/12)*b*log(b^2+1)*(1-b^2)+(1/24)*arctan(1/b)*(1-6*b^2+b^4) 
elif n = 6 then (1/4)*delta^4-(1/3)*delta^2*b+(1/12)*delta^2*(3*b^2-1)+(1/6)*b*delta*(1-b^2)+(137/1440)*b^4-(137/720)*b^2+137/7200+(1/240)*log(b^2+1)*(5*b^4-10*b^2+1)+(1/120)*b*arctan(1/b)*(-5+10*b^2-b^4) 
elif n = 7 then (1/5)*delta^5-(1/4)*delta^4*b+(1/18)*delta^3*(3*b^2-1)+(1/12)*delta^2*b*(1-b^2)+(1/120)*delta*(5*b^4-10*b^2+1)-(49/2400)*b+(49/720)*b^3-(49/2400)*b^5+(1/720)*b*log(b^2+1)*(-3*b^4+10*b^2-3)+(1/720)*arctan(1/b)*(-1+15*b^2-15*b^4+b^6) 
end if 
end proc:

plot(Y(5, b, 7), b = 0 .. 10); # Example of use

 

Use a correct syntax....

Mariusz Iwaniuk 1571
May 05 2018
0 1

u1 := proc (x, y) options operator, arrow; 1-exp(a*x)*cos(2*Pi*y) end proc;

a := -0.39323780;

evalf(int(u1(x, y)^2, y = -.5 .. 1.5, x = -.5 .. 1.5));

#5.493248990

#OR:

u := unapply(1-exp(a*x)*cos(2*Pi*y), [x, y]);

a := -0.39323780;

evalf(int(u(x, y)^2, y = -.5 .. 1.5, x = -.5 .. 1.5));

#OR:

u := 1-exp(a*x)*cos(2*Pi*y);

a := -0.39323780;

evalf(int(u^2, y = -.5 .. 1.5, x = -.5 .. 1.5));

....

Mariusz Iwaniuk 1571
May 05 2018
0 0

Use  ExcelTools for import file.

 


 

Download testxlsx.mw

 

....

Mariusz Iwaniuk 1571
May 02 2018
0 0

I don't have Maple V (This is an obsolete version,you need upgrade).

integral := Int(2*(sin(theta)/cos(theta))^(2*p-1), theta = 0 .. (1/2)*Pi);

with(IntegrationTools):

integral2 := Change(convert(integral, tan), tan(theta) = sqrt(t));

`assuming`([value(integral2)], [0 < p and p < 1]);

#Answer is: Pi*csc(Pi*p)

integral.mw

Executed in Maple 2018

Using build-in function pdetest....

Mariusz Iwaniuk 1571
April 30 2018
1 2

Maple can't simplify better, probably V is not a solution of pde[1].

See attached file:

solution_ver_3.mw

In Maple 2018....

Mariusz Iwaniuk 1571
April 30 2018
0 2

BVP := [4*(diff(u(x, t), t))-9*(diff(u(x, t), x, x))-5*u(x, t) = 0, u(0, t) = 0, u(6, t) = 0, u(x, 0) = sin((1/6)*Pi*x)^2];

sol := pdsolve(BVP);

plot3d(eval(rhs(sol), infinity = 10), x = 0 .. 6, t = 0 .. 4);

OR:

plot3d(subs(infinity = 10, rhs(sol)), x = 0 .. 6, t = 0 .. 4);

OR:

plot3d(subs(infinity = 10, op(2, sol)), x = 0 .. 6, t = 0 .. 4)

....

Mariusz Iwaniuk 1571
April 30 2018
3 0

 I am no expert in special functions,but adding comand "_EnvLegendreCut"

_EnvLegendreCut := 1 .. infinity;

plot(LegendreQ((1/2)*sqrt(5)-1/2, x), x = -1 .. 1);

for more info execute command:

?LegendreQ;

seq,ListTools,select.....

Mariusz Iwaniuk 1571
April 29 2018
1 0

restart;

with(ListTools):

l := seq(n, n = -10 .. 20);

[l];

l1 := Flatten([Transpose([[seq(n, n = -10 .. 20)], [seq(n, n = 1 .. 31)]])]);

l2 := Transpose([[seq(n, n = -10 .. 20)], [seq(n, n = 1 .. 31)]]);

l3 := select(type, [seq(n, n = 115 .. 231)], 'odd');

[l3];

BUG...

Mariusz Iwaniuk 1571
April 27 2018
0 9

In yours first example Maple gives incorect answer.It's a Bug.!!!

 

evalf[10](sum(2^n*floor(2^n), n = 1 .. infinity));

#-1.333333333

evalf[10](Sum(2^n*floor(2^n), n = 1 .. infinity));# Big 'S' in Sum

#-1.333333333

`assuming`([sum(2^n*floor(2^n), n = 1 .. m)], [m > 0]);

#(1/3)*2^(m+1)*floor(2^(m+1))-4/3

limit(%, m = infinity)

#infinity

Second one give a correct answer:

evalf[10](sum(5^(n-1)*floor((1/4)*5^n), n = 1 .. infinity));

#Float(infinity)

evalf[10](Sum(5^(n-1)*floor((1/4)*5^n), n = 1 .. infinity));# Big 'S' in Sum

#If Maple dosen't know the answer then: Returns unevaluated,not infinity in this case.

`assuming`([sum(5^(n-1)*floor((1/4)*5^n), n = 1 .. m)], [m > 0]);

#Returns unevaluated

 

Executed in Maple 2018.

CauchyPrincipalValue....

Mariusz Iwaniuk 1571
April 26 2018
3 2

Using CauchyPrincipalValue =true:

`assuming`([int(exp(I*x*t)/((x-a)*(x+a)), x = -infinity .. infinity, CauchyPrincipalValue = true)], [a > 0, t > 0])

#(1/2*I)*Pi*(exp(I*a*t)-exp(-I*a*t))/a

int.mw

Exectuted in Maple 2018.

Another way...

Mariusz Iwaniuk 1571
April 25 2018
0 0

remove[flatten](x-> x = 0, [seq(sin((1/4)*k*Pi), k = 1 .. 8)]);

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