Section 11.10: Taylor Series

Functions as power series

Questions

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

What is the domain of this function?

What is k(0)?

Write out the first 5 terms of k(x), the first 5 terms of k'(x) and the first 5 terms of ∫k(x)d x. What well known function is this?

Taylor series

Given a function we want to find a power representation for it if possible.

We need to specify a center of convergence (a MacLaurin series is just a Taylor series centered at 0).

A good power representation is one for which it converges for more than just the center.

Building a power series from a known series, like the geometric series, works in some but not all cases.

Questions

Given function f(x), suppose it has a power series representation centered at x=a.
	f(x)=(Underoverscript[Σ, (n=0), ∞])(((Subscript[c, n])(x-a))^n)
What are the values (Subscript[c, n])?

Write this function with the first 6 terms of the power series written out (n=0,1,2,3,4,5).

What happens when x=a is plugged into this function?

What happens when x=a is plugged into this function after both sides are differentiated.

What happens when x=a is plugged into this function after both sides are differentiated twice.

What happens when x=a is plugged into this function after both sides are differentiated three times.

What happens when x=a is plugged into this function after both sides are differentiated four times.

In general, what appears to be true?

Definition

If f(x) is differentiable to all orders at x=a, then its Taylor series centered at a is
	(Underoverscript[Σ, (n=0), ∞])(((f^((n)))(a))/(n!))(((x-a))^n)

From the above, if f(x) has a power series representation centered at a, it must be its Taylor series.

Clear[f,a]
Series[f[x],{x,a,5}]

Comment:

The tangent line (the osculating line) to the graph at x=a is best written -- and best thought of, perhaps -- as (Subscript[S, 1]). Write that out, and confirm it.

What would you call (Subscript[S, 2])?

What would you call (Subscript[S, 3]), etc.?

Questions

What is the Taylor series centered at 0 for f(x)=(1/(1-x))?

eff[x_]:=1/(1-x)
Series[eff[x],{x,0,10}]
deff[x_]=D[eff[x],{x,100}]
deff[0]
100!

What is (f^((100)))(0)?

What is (f^((501)))(0)?

Questions

What is the Taylor series centered at 0 for f(x)=ln(1+x)?

eff[x_]:=Log[1+x]
Series[eff[x],{x,0,10}]
deff[x_]=D[eff[x],{x,100}]
deff[0]
-99!

What is (f^((100)))(0)?

What is (f^((501)))(0)?

What is the Taylor series centered at 0 for x ln(1+x)?

Questions

Let f(x)=(e^x).

What is the Taylor series for f(x) centered at 0?

Series[Exp[x],{x,0,10}]

What is (f^((n)))(x)?

What is (f^((n)))(0)?

For which values of x does this series converge?

Using the above, what is the Taylor series for k(x)=x (e^x) centered at 0?

What is true about (k^((n)))(0)?

Questions

Let g(x)=sin(x).

Series[Sin[x],{x,0,10}]

What is the Taylor series for g(x) centered at 0?

What is (g^((n)))(x)?

What is (g^((n)))(0)?

For which values of x does this series converge?

Using the above, what is the Taylor series for m(x)=(x^2) sin(x) centered at 0?

What is true about (m^((n)))(0)?

Questions

Let h(x)=cos(x).

Series[Cos[x],{x,0,10}]

What is the Taylor series for h(x) centered at 0?

For which values of x does this series converge?

What is (lim_(x->0))((1-cos(x))/(x^2))?

Using the above, what is the Taylor series centered at 0 for ((1-cos(x))/(x^2))?

Questions

Let h(x)=cos(x).

Series[Cos[x],{x,π/2,10}]
Series[-Sin[x-π/2],{x,π/2,10}]

What is the Taylor series for h(x) centered at π/2?

For which values of x does this series converge?

Questions

Let k(x)=cos((x^2)).

Series[Cos[x^2],{x,0,10}]

Using the Taylor series centered at 0 for cos(x), what is the Taylor series centered at 0 for k(x)?

The derivative (k^((n)))(0) is 0 for which values of n?