| The ebook Elementary Calculus is based on material originally written by H.J. Keisler. For more information please read the copyright pages. |
|
Home  Infinite Series Taylor Series Theorem 1 |
|||||||||||
 Search the VIAS Library | Index
|
  |
||||||||||
Theorem 1
Theorem 1 shows that if we already know a function f(x) is the sum of a power series, then that power series must be the Taylor series of f(x). THEOREM 1 Suppose f(x) is equal to the sum of a power series with radius of convergence r > 0,
Then the power series is the same as the Taylor series for f about c. In other words, an is just f(n)(c)/n! for n = 0, 1, 2, ... Discussion A function which is equal to the sum of a power series in x - c (with nonzero radius of convergence) is called analytic at c. The theorem shows that every analytic function is equal to the sum of its Taylor series. PROOF Since power series can be differentiated term by term within its interval of convergence, all the nth derivatives f(n)(c) exist. Let us compute f(n)(x) and set x = c.
Thus for each n, an= f(n)(c)/n!, and the original power series is the same as the Taylor series of f(x),
|
|||||||||||
| Home Infinite Series Taylor Series Theorem 1 |
|
||||||||||
Last Update: 2006-11-08