Whatever is the reason,

plots:-display(seq(plo[i], i = 1 .. 30))

instead of with(plots); display(seq([plo[i]], i = 1 .. 30)) seems to be working.

Alec

Hi Paulina,

Did each of you buy a book, or there was one that you shared?

I tried to find a secret download link in your post - something like "Help to these people..." which means http, but unsuccessfully.

Alec

Actually, these BellB polynomials also may be useful for sin(cos(x)) - expressing sin and cos through exponents and using the following formula:

diff(exp(exp(x)),x$n)=BellB(n,exp(x))*exp(exp(x))

Alec

In the first case, the closed form may not exist. In the second case, the radius of convergence is 0 which means that the series is divergent everywhere except x=-7 where it is 1.

Also, there should be a multiplication sign between parentheses, i.e.

(2*n+1)*(x+7)

If you meant to use n-th power only for (x+7), then Maple would produce the answer,

sum((2*n+1)*(x+7)^n,n=0..infinity);

                                  x + 8
                            g := --------
                                        2
                                 (x + 6)

Alec

Increase printlevel - set it as 

printlevel:=2;

Without that, print command also can be used for plots.

Alec

There are 2 different objects called Bell polynomials

The polynomials of one variable can be defined as

BellB:=(n,x)->add(combinat:-stirling2(n,k)*x^k,k=0..n);

Their values at 1 are Bell numbers. For example,

BellB(10,1)=combinat:-bell(10);

                           115975 = 115975

Alec

For example,

f:=(x,y)->x^2+y:
g:=(x,y)->x^2+y-sin(y):
h:=x->f(x,fsolve(g(x,y),y)):
plot(h,-1..1);

Alec

Mathematica gives the following answer in case a=1 (the general case can not be reduced to that),

(1+3*b^2)*Pi*((1+b^2)*cos(2*p)+2*I*b*sin(2*p))/(-1+b^2)^2

Alec

A simple series for the solution can be obtained as

eq:=v*t=(S0-St)+Km*ln(S0/St): 
s:=solve(eq,St):
alias(t0=(-Km+ln(S0)*Km-ln(Km)*Km+S0)/v):
simplify(series(s,t=t0));
                        2                  3                    4
                       v            2     v             3      v
  Km - v/2 (t - t0) + ----- (t - t0)  + ------- (t - t0)  - --------
                      16 Km                   2                    3
                                        192 Km              3072 Km

                          5
                4     13 v            5             6
        (t - t0)  - --------- (t - t0)  + O((t - t0) )
                            4
                    61440 Km

Alec

The series for LambertW(x) is

sum((-n)^(n-1)*x^n/n!,n=1..infinity);

                       infinity
                        -----       (n - 1)  n
                         \      (-n)        x
                          )     --------------
                         /            n!
                        -----
                        n = 1

It's radius of convergence is

limit(1/(n^(n-1)/n!)^(1/n),n=infinity);

                               exp(-1)

However, evalf/sum in Maple works even for much larger values. For example,

evalf(eval(%%,x=10));

                             1.745528003

LambertW(10.);

                             1.745528003

By the way, this series, together with the radius of convergence, can be found in Maple as

FunctionAdvisor(LambertW,sum_form);

                 infinity
                  -----         (_k1 - 1)  _k1
                   \      (-_k1)          z                   1
  [LambertW(z) =    )     --------------------, And(| z | < ------)]
                   /              _k1!                      exp(1)
                  -----
                 _k1 = 1

Alec

neither worked for me for the power 10000 though, instead of 2000, and the Interrupt button couldn't even be pushed, because the cursor over it became a rotating wheel.

Also, I couldn't stop Maple by pushing the x button in the top right corner - had to use Task Manager to kill it.

Alec

Oh, I looked at the wrong integral - without abs.

Alec

add something like

assuming r>R, R>a

just in front of ; at the end. 

Alec

That looks like a bug - probably, the values were not properly casted in the sort code, in which case the compiler gives just a warning, not an error, and that warning was ignored.

Alec

f:=sum(x^(2*n)/(n+1)!,n=0..infinity);
                                     2
                                exp(x ) - 1
                           f := -----------
                                     2
                                    x

g diverges for every x except x=-5, in which case it is 0, so does f*g.

Alec