1. Power Series Salas, Hille, Etgen Calculus: One and Several Variables
Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

2. Power Series There are exactly three possibilities for a power series:
Case 1. The series converges only at x = 0. This is what happens with
For x ≠ 0, kk xk , and so the series cannot converge.
Case 2. The series converges absolutely at all real numbers x. This is what happens with the exponential series
Case 3. There exists a positive number r such that the series converges absolutely for |x| < r and diverges for |x| > r . This is what happens with the geometric series
Here there is absolute convergence for |x| < 1 and divergence for |x| > 1.
Salas, Hille, Etgen Calculus: One and Several Variables
Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

3. Power Series Associated with each case is a radius of convergence:
In Case 1, we say that the radius of convergence is 0.
In Case 2, we say that the radius of convergence is ∞.
In Case 3, we say that the radius of convergence is r.
Salas, Hille, Etgen Calculus: One and Several Variables
Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

4. Differentiation & Integration of Power Series
Salas, Hille, Etgen Calculus: One and Several Variables
Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

5. Differentiation & Integration of Power Series
Salas, Hille, Etgen Calculus: One and Several Variables
Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

6. Differentiation & Integration of Power Series
Salas, Hille, Etgen Calculus: One and Several Variables
Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

7. Differentiation & Integration of Power Series
Term-by-term integration can be expressed by writing
Salas, Hille, Etgen Calculus: One and Several Variables
Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

8. Differentiation & Integration of Power Series
Power Series; Taylor Series
It is time to relate Taylor series
to power series in general. The relationship is very simple.
Salas, Hille, Etgen Calculus: One and Several Variables
Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.