北京邮电大学国际学院高等数学(下)幻灯片讲义(无穷级数)Lecture 2

Convergence of Power Series

n

n n a x ∞

=∑0x 00x ≠0||||x x <0x 00x ≠0||||x x >Abel’s Theorem Consider the series .(1)

If it converges at , , then it must converge absolutely in the interval ;(2)

If it diverges at , , then it must diverge in .

9the sequence is bounded, that is, such that

0{}n

n a x 0M ∃>Proof of Part (1)

Since converges, .01

n

n n a x ∞

=∑0lim 0n

n n a x →∞

=Thus 0

||N .

n

n a x M

n +≤∈for all

A Function f whose Taylor Series Converges at every x but Converges to f (x )only at x = 0

It can be shown ( although not easily ) that

37

has derivatives of all orders at x = 0and that for all n .

()

(0)0n f =Then the question is:

For what values of normally expect a T to converge to its generating function?

The Series of Functions

Power Series

The Convergence of Power Series Taylor and Maclaurin Series Combining Taylor Series