Convergence

If you have a sequence of values, for which

> lim(alpha[n],n=1..infinity)=Gamma;

then we say that converges to . The convergence is of order if there exist a number independent of , for which

> abs((alpha[n]-Gamma)/g[n]) <= K;

For example, with our data from above:

> for n from 1 to 6 do alpha[n]:=subs(x=0.6,sum(x^i/(i!),i=0..n)): print (n,abs(p-alpha[n]), 2*0.6^(n+1)/(n+1)!); od;

>

This shows that the convergence is of order

.

> series(exp(x),x,5);

which illustrates the order of the error term. Since the term makes the right hand side in the inequality smaller, one can also omit it and simply say that convergence is as indicated in the output to the MapleV 'series()' command.