kill(all)$

maxima :intro  powerseries, version  1.0 
peter.vlasschaert@gmail.com,13/11/2020

youtube :https://www.youtube.com/watch?v=V1AKAkGJlN8

load packages ,option
===================

load(lrats)$ ratprint:false$

given
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(1+x)^k = 1+k*x/1! +k*(k-1)*x^2/2! +  ........+ k*(k-1)*.......(k-(j-1))*x^j+........

q1:product(k-j,j,0,4);

q2:1+sum(product(k-j,j,0,l-1)*x^l/(l)!,l,1,m);

q2a:subst([m=3],q2);

q2b:ev(q2a,nouns);

powerdisp:true;/*low power to high power*/

q2c:q2b;

power serie : (1+x)^k

q3:(1+x)^k = subst([m=inf],q2);

→  not power serie , m≠ inf ( = approximation)

q3a:lhs(q3) = q2;

find : 1/(1+x)^2
---------------------

q4:subst([k=-2],q3);

approximation : some of the terms,of the power serie.
===========

q4a:lhs(q4)=sublis( [m=3,k=-2],rhs(q3a));

ratsimp(1+sum(((product(-2-j,j,0,-1+l))*x^l)/l!,l,1,3));

q4b:lhs(q4a)=ev(rhs(q4a),nouns);

→ power serie ,from q4b,alternating power serie.

q4c:lhs(q4b) =limit( sum((i+1)*x^i*(-1)^i,i,0,m),m,inf);

q4c1:lhs(q4c) =subst([m=inf], args(rhs(q4c))[1]);

→ how get this from menu :  Calculus →get series

series:

1e) Expression : 1/(1+x)^2
2e) Variable      : x
3e) Select :power series

niceindices(powerseries(1/(1+x)^2, x, 0));

q5:lhs(q4c) = %;

relation :between : q4c1 and q5
--------------------------------------------

is(rhs(q4c1)=rhs(q5));

find : 1/(2+x)^3
---------------------

q6a:1/(2+x)^3;

q6b:q6a = 1/(a*(1+x/2)^3);

q6c:solve(q6b,a);

q6d:subst(q6c,q6b);

q6d1:part(denom(rhs(q6d)),1);

→ use: q3a

q6e:q3a;

→ replace x => x/2 , k = -3

q6e1:subst([x=x/2,k=-3],q6e);

→ multiplication 1/q6d1

q6e2: 1/q6d1*q6e1;

q6e3:lhs(q6e2)=expand(rhs(q6e2));

→ approximation :some terms ,m =3

q6e4:lhs(q6e3) = subst([m=3],rhs(q6e3));

q6e5:lhs(q6e3)= ev(rhs(q6e4),nouns);

→ expand 

q6e6:lhs(q6e5)=expand(rhs(q6e5));

→ power serie  : m →∞,use  q6e2

q6e7:lhs(q6e2) = expand(subst([m=inf],rhs(q6e2)));

→ how get this from menu :  Calculus →get series

series:

1e) Expression : 1/(2+x)^2
2e) Variable      : x
3e) Select :power series

niceindices(powerseries(1/(2+x)^3, x, 0));

q6e8:lhs(q6e7)= %;

q6e9:args(rhs(q6e8));

→ approximation :  i = 0 ,1 , ...,m (=3)

q6e10:lhs(q6e8) = sum(q6e9[1],i,0,3);

→ check, between 'q6e10 , q6e6'

q6e11:is(rhs(q6e10)= rhs(q6e6));

more info : youtube.for this example

find :(1+x)^(1/2) , example : fractional power = 1/2 (= k)
---------------------

use :q3a

q7e1:q3a;

use : k = 1/2, m = 3

q7e2:subst([k=1/2,m=3],q7e1);

→ expand 

q7e3:lhs(q7e2)=ev(rhs(q7e2),nouns);

→ power serie ( k = 1/2,m=∞)

q7e4:subst([k=1/2,m=inf],q7e1);

→ how get this from menu :  Calculus →get series

series:

1e) Expression : (1+x)^(1/2)
2e) Variable      : x
3e) Select :power series

niceindices(powerseries((1+x)^(1/2), x, 0));

q7e5:lhs(q7e4) = %;

→ how get this from menu :  Calculus →get series

series: ' taylor series'

1e) Expression : (1+x)^(1/2)
2e) Variable      : x
3e) point : 0
4e) depth : 3

q7e6:taylor((1+x)^(1/2), x, 0, 3);

fourth : element of the power serie.

q7e7:part(trunc(q7e6),4);

approximation :sqrt(1+x)

q7e8:sqrt(1+x)=1+(2*sum(x^i/beta(3/2-i,1+i),i,1,3))/3;

q7e9:expand(q7e8);

→ what about :beta(3/2-i,1+i), representation ; fractional order ( k =1/2,q7e1)

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     ref :http://maxima.sourceforge.net/docs/manual/maxima_81.html
     ref :https://web.csulb.edu/~woollett/mbe7sint.pdf

Function: beta (a, b)
----------------------------

    The beta function is defined as gamma(a) gamma(b)/gamma(a+b) (A&S 6.2.1).
============================================================== 

Function: gamma (z)
-----------------------------

    The basic definition of the gamma function (A&S 6.1.1) is 
==============================================================
→  :gamma(z)=integrate(t^z*exp(-t),t,0,inf) ,z = a,b
relation :Im(z)=0  ,Re(z) = n ,z =complex number = x+%i*y  , %i =sqrt(-1)
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n = positive integer and 0 = {0,1,2,...........}
→   n! = gamma(n+1)
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