13) Series and Sequences
We can find the sum of convergent series.
Consider the geometric series with a=1 and r =1/2. Since |r|<1, it converges, and the sum is
> restart;
>
Consider the p-series with p=2. Since p>1, it converges, and the sum is given by
>
Taylor Series
Mclaurin polynomial of sin(x) near x =0.
> restart;
> with(plots):
> f:=cos(x);
> taylor( f, x=0,10);
they can be ploted sequentially
> for n from 1 by 2 to 10 do
> p[n]:=mtaylor(f,[x=0],n);
> plot({f,p[n]},x=-3..3);
> od;
they can be ploted together
> plot({f,seq(p[n],n=1..5)},x=-3..3,y=-1..1);
Taylor polynomial of e^x near x =1.
> f:=exp(x);
they can be ploted sequentially
> for n from 1 by 1 to 5 do
> p[n]:=mtaylor(f,[x=1],n);
> plot({f,p[n]},x=-3..3);
> od;
they can be ploted together
> plot({f(x),seq(p[n],n=1..5)},x=-1..2);