Section 11.8: Power Series

Review

Two tests that will be the workhorses for our analysis are based on comparisons to geometric series.

Ratio test

Given any series (Underoverscript[Σ, k, ∞])(Subscript[b, k]), evaluate the limit of the ratio of consecutive terms, ignoring any signs, (lim_(k->∞))|((Subscript[b, k+1])/(Subscript[b, k]))|.

If (lim_(k->∞))|((Subscript[b, k+1])/(Subscript[b, k]))|=L where L<1, then (Underoverscript[Σ, k, ∞])(Subscript[b, k]) converges absolutely.

If (lim_(k->∞))|((Subscript[b, k+1])/(Subscript[b, k]))|=L where L>1, then (Underoverscript[Σ, k, ∞])(Subscript[b, k]) diverges.

If (lim_(k->∞))|((Subscript[b, k+1])/(Subscript[b, k]))|=L where L=1, then you must use another convergence test.

Root test

Given any series (Underoverscript[Σ, k, ∞])(Subscript[b, k]), evaluate the limit of the (k^th) root, ignoring any signs, of the (k^th) term, (lim_(k->∞))((|(Subscript[b, k])|)^(1/k)).

If (lim_(k->∞))((|(Subscript[b, k])|)^(1/k))=L where L<1, then (Underoverscript[Σ, k, ∞])(Subscript[b, k]) converges absolutely.

If (lim_(k->∞))((|(Subscript[b, k])|)^(1/k))=L where L>1, then (Underoverscript[Σ, k, ∞])(Subscript[b, k]) diverges.

If (lim_(k->∞))((|(Subscript[b, k])|)^(1/k))=L where L=1, then you must use another convergence test.

Power series

How can we approximate sin(1) or (e^(-1.5))? We can represent both of these as power series, which are essentially series with polynomials as terms rather than just constants. So they’re functions, written as infinite sums of “monomials”. I hope that you’re thinking “Cool!” -- but you might be thinking “Oh my God!”...:)

Turns out that a lot of our familiar functions can be written this way; furthermore, we can then approximate particular values of the functions using partial sums. In particular, many transcendental functions, like sin(x) and (e^x) can be written as power series in x: polynomials of infinite degree.

Definition

A power series in x centered at a is a series of the form
	(Underoverscript[Σ, (n=0), ∞])(((Subscript[c, n])(x-a))^n)
where (Subscript[c, n]) is an expression in n and a is a number.

Questions

Identify the (Subscript[c, n]) and a for each of the following power series.

(((Underoverscript[Σ, (n=2), ∞])(-1))^n)((((x-1))^n)/n)

(Underoverscript[Σ, (n=1), ∞])(((2n-1))/(n^2))(x^n)

(((Underoverscript[Σ, (n=0), ∞])(3x))^n)

(Underoverscript[Σ, (n=0), ∞])((((5x-3))^n)/(n!))

Questions

A function is defined as the power series
	f(x)=(Underoverscript[Σ, (n=0), ∞])((((x-1))^n)/(2^n))=1+((x-1)/2)+((((x-1))^2)/4)+((((x-1))^3)/8)+…

What is f(1)?

What is f(2)?

Is f(3) defined? If so, what is it? If not, why not?

To determine the domain of this function, what do we need to worry about?

Terminology

The interval of convergence is the domain for a power series.

The radius of convergence is the distance from the center to the edge of the interval of convergence. It comes from the ratio/root test.

Questions

A function is defined as the power series
	g(x)=(Underoverscript[Σ, (n=1), ∞])((3(((x+4))^n))/n)

What is the center of this series?

For what value of x is it easy to evaluate this function?

What is the radius of convergence for this power series?

What is the interval of convergence for this power series?

Questions

Let H(x)=(Underoverscript[Σ, (n=0), ∞])((x^n)/(n!)).

What are the first 4 terms of the series?

What is the center of this series?

What is the radius of convergence for this power series?

What is the interval of convergence for this power series?

Using an appropriate error estimate approximate H(-1) with error less than 0.0001.

Factorial[8]>10000
Factorial[7]>10000
approx=N[Sum[(-1)^n/Factorial[n],{n,0,7}]]
exact=Exp[-1]
Abs[approx-exact]

(* Note: Someone wondered about plotting factorials. 
They're really only defined for positive integers, but the Gamma function is a "special function" that extends the factorials: *)
p1=Plot[Gamma[x+1],{x,0,4}];
p2=ListPlot[Table[{i,Factorial[i]},{i,0,4}]];
Show[p1,p2]
p1=Plot[{Gamma[x+1],10000},{x,0,8}];
p2=ListPlot[Table[{i,Factorial[i]},{i,0,8}]];
Show[p1,p2]

What is this function? Let’s plot some partial sums....

S10[x_]:=Sum[x^n/Factorial[n],{n,0,10}]
Plot[S10[x],{x,-2,2},PlotRange->{0,6}]

Can you guess the function?

Plot[{Exp[x],S10[x]},{x,-2,2},PlotRange->{0,6},PlotLabel->"Can't really tell them apart (yet)"]
(* let's look at the error: *)
Plot[Exp[x]-S10[x],{x,-2,2},PlotRange->All,PlotRange->All,PlotLabel->"Error looks pretty good for (-2,2):"](* let's watch S10 break down: S10 is just a 10th degree polynomial, and it won't work for long! *)
Plot[{Exp[x],S10[x]},{x,-10,5},PlotRange->All,PlotLabel->"Whoops! Break down! Exponentials don't do that...."]

Questions

Let F(x)=(Underoverscript[Σ, (n=0), ∞])((((2x+3))^n)/((n^2)+1)).

What are the first 4 terms of the series?

What is the center of this series?

What is the radius of convergence for this power series?

What is the interval of convergence for this power series?

Using an appropriate error estimate approximate F(-1) with error less than 0.0001.

Clear[n]
true=Sum[1/(n^2+1),{n,0,Infinity}]
(* Compute N in two ways: comparison, with p-series: *)
soln=Solve[Integrate[1/x^2,{x,n,Infinity}]==.0001,Reals]
enn=n/.soln
enn=Ceiling[enn]
approx=N[Sum[1/(n^2+1),{n,0,enn}]]
Abs[approx-true]
(* Direct, integral test: *)
Integrate[1/(1+x^2),{x,n,Infinity}]
soln=Solve[Pi/2-ArcTan[n]==.0001,n,Reals]
enn=n/.soln
enn=Ceiling[enn]
approx=N[Sum[1/(n^2+1),{n,0,enn}]]
Abs[approx-true]