Convergence of Power Seriesrestart:A power series NiMtJSJQRzYjJSJ4Rw== in NiMlInhH is a sum of constants times powers of NiMlInhH:NiMtJSJQRzYjJSJ4Rw== = NiMmJSJDRzYjIiIh + NiMqJiYlIkNHNiMiIiJGJyUieEdGJw== + NiMqJiYlIkNHNiMiIiMiIiIqJCklInhHRidGKEYo + ... + NiMqJiYlIkNHNiMlIm5HIiIiKSUieEdGJ0Yo + ... = NiMtJSRzdW1HNiQqJiYlIkNHNiMlIm5HIiIiKSUieEdGKkYrL0YqOyIiISUpaW5maW5pdHlH.An example is the Taylor series for NiMvLSUiZkc2IyUieEctJSRzaW5HRiY= about NiMvJSJ4RyIiIQ==:NiMtJSJURzYjJSJ4Rw== = NiMsKiUieEciIiIqJilGJCIiJEYlLSUqZmFjdG9yaWFsRzYjRighIiJGLComKUYkIiImRiUtRio2I0YvRixGJSomKUYkIiIoRiUtRio2I0Y0RixGLA== + ... + NiMqKCksJCIiIiEiIiUibkdGJiklInhHLCYqJiIiI0YmRihGJkYmRiZGJkYmLSUqZmFjdG9yaWFsRzYjRitGJw== + ...The first question of interest for a power series NiMtJSJQRzYjJSJ4Rw== is "for what values of NiMlInhH does the power series converge?" Secondly, if the power series is a Taylor series NiMtJSJURzYjJSJ4Rw== for NiMtJSJmRzYjJSJ4Rw== and converges for a given NiMlInhH, does NiMvLSUiVEc2IyUieEctJSJmR0Ym? f:=sin(x);Next we calculate and plot in succession the first 31 Taylor polynomials about NiMvJSJ4RyIiIQ==.for n from 0 to 30 do
P[n]:=convert(taylor(f,x=0,n+1),polynom):
od:for n from 0 to 30 do
plot({f,P[n]},x=-20..20,y=-10..10,color=[red,blue],thickness=3):
od;We notice that the higher the degree of the polynomial, the farther we can move away from 0 and still have a good approximation. Next we look at all the graphs together.plot({f,seq(P[n],n=0..30)},x=-20..20,y=-10..10,thickness=3);What does it mean for a power series to converge for a given NiMlInhH? For NiMvJSJ4RyIiJg==, we look at the sequence NiMtJiUiUEc2IyUibkc2IyIiJg== for NiMlIm5H from 0 to 40.P:='P':
for n from 0 to 40 do
P[n](5)=evalf(subs(x=5,convert(taylor(f,x=0,n+1),polynom)));
od;We see that this sequence settles in on one particular number as NiMlIm5H gets "large." Just to check if it stays at that number for even larger NiMlIm5H, let's find NiMtJiUiUEc2IyIkKyM2IyIiJg==.evalf(subs(x=5,convert(taylor(f,x=0,201),polynom)));We take it that the sequence of partial sums converges to -.9589242747 to 10 decimal places, and take this limit to be NiMtJSJURzYjIiIm. but is this the same as NiMtJSRzaW5HNiMiIiY=?evalf(sin(5));It seems so. In general, for any power series NiMtJSJQRzYjJSJ4Rw== and any given NiMvJSJ4RyUiYUc=, we say that NiMtJSJQRzYjJSJ4Rw== converges to NiMlIkxH for NiMvJSJ4RyUiYUc= if NiMvLSUmbGltaXRHNiQtJiUiUEc2IyUibkc2IyUiYUcvRislKWluZmluaXR5RyUiTEc=. Now, if NiMtJSJQRzYjJSJ4Rw== is a Taylor series for the functions NiMtJSJmRzYjJSJ4Rw== we will be considering in this course (but not necessarily other functions), NiMvLSUiZkc2IyUiYUclIkxH also, i.e., the Taylor series converges to the function value at the given point NiMlImFH. We note here that the Taylor series for NiMtJSRleHBHNiMlInhH, NiMtJSRzaW5HNiMlInhH, and NiMtJSRjb3NHNiMlInhH converge for all NiMlInhH. Next let's consider the function NiMvLSUiZkc2IyUieEcqJiIiIkYpLCZGKUYpRichIiJGKw== .restart:f:=1/(1-x);This function has as its Taylor seriesNiMvLSUiVEc2IyUieEcsLCIiIkYpRidGKSokKUYnIiIjRilGKSokKUYnIiIkRilGKSokKUYnIiIlRilGKQ== + ... + NiMpJSJ4RyUibkc= + ...We calculate and plot in succession the first 11 Taylor polynomials about NiMvJSJ4RyIiIQ==.for n from 0 to 10 do
P[n]:=convert(taylor(f,x=0,n+1),polynom):
od:for n from 0 to 10 do
plot({f,P[n]},x=-2..2,y=-10..10,color=[red,blue],thickness=3,discont=true):
od;We see that our approximations are good between -1 and 1, but nowhere else. In fact, all ot the Taylor polynomials are positive to the right of 1, but NiMtJSJmRzYjJSJ4Rw== is negative there. Let's look at all of the graphs together.plot({f,seq(P[n],n=0..10)},x=-2..2,y=-10..10,thickness=3,discont=true);We have seen that some power series converge for all NiMlInhH. Others, such as this one, converge exactly for all NiMlInhH within some radius NiMlIlJH of the center of the series, called the radius of convergence of the series. A final type of series converges only at a single point, its center. For our current series, let's illustrate its convergence at NiMvJSJ4RyQiIiohIiI=. We can easily calculate that NiMvLSUiZkc2IyQiIiohIiIiIzU=.P:='P':
for n from 0 to 220 do
P[n](.9)=evalf(subs(x=.9,convert(taylor(f,x=0,n+1),polynom)));
od;This certainly appears to be a convergent sequence, so we conclude the series converges for NiMvJSJ4RyQiIiohIiI=. Now let's try the same for NiMvJSJ4RywkIiIiISIi and NiMvJSJ4RyIiIg==.for n from 0 to 50 do
P[n](-1)=evalf(subs(x=-1,convert(taylor(f,x=0,n+1),polynom)));
od;for n from 0 to 50 do
P[n](1)=evalf(subs(x=1,convert(taylor(f,x=0,n+1),polynom)));
od;Quite clearly, there is no convergence here. The radius of convergence for NiMvLSUiZkc2IyUieEcqJiIiIkYpLCZGKUYpRichIiJGKw== is 1, which tells us that the series converges for NiMlInhH between -1 and 1. But one may or may not have convergence at the end points of the interval of convergence. Here we don't. Finally, let's look at -1.01, which is outside of the interval of convergence. We first find NiMtJSJmRzYjLCQkIiQsIiEiIyEiIg==.1/(1-(-1.01));for n from 0 to 100 do
P[n](-1.01)=evalf(subs(x=-1.01,convert(taylor(f,x=0,n+1),polynom)));
od;We have no hint of convergence here.