講者 : 許永昌 老師 1

Contents Sequences of functions Uniform convergence Weierstrass M test Abel’s test Taylor’s Expansion Remainder Radius of convergence Binomial Theorem Power Series Properties Inversion of power series 2

Sequences of functions ( 請預讀 P276~P277) 3

Uniform Convergence ( 請預讀 P277) Example: f n (x)=sin x + 1/n  cos5x We can find that f(x)=sin x. |f n (x)-f(x)| 5. |f n (x)-f(x)| N. 4 

Useful properties of Uniformly Convergent Sequence 5

Example of nonuniform convergence ( 請預讀 P277) Case I: f n (x)=sin n (x). Case II: s(0)=0, s(x  0)=1 It has a discontinuity at x=0. However, {s n (x)} are continuous functions for all finite n. It is not a uniformly convergent series in the interval [0,1]. 6 n increases x sn(x)sn(x) 

Weierstrass M (Majorant) Test 7 M3M3 M2M2 M1M1 x un(x)un(x) x sn(x)sn(x) 此圖例的  M i 不收斂

Uniform and absolute convergence The uniform convergence and absolute convergence are independent properties. 8 YesNo Yes No Absolute Uniform

Abel’s Test ( 請預讀 P279) 9

Useful properties of a uniformly convergent series 10

Homework

Taylor’s Expansion 12 *a=0 for Maclaurin series.

The Remainder term of a Taylor’s expansion ( 請預讀 P281~P282) We get 13

The Remainder (continue) If g(x) is a continuous function in [a,b], any  [g min,g max ] can find at least one  in which M=g(  ),  [a,b]. This formula is important for numerical calculation. 14

Example where We get the error of Taylor’s expansion for numerical calculation. 15 x

Examples: find the convergent range of a Taylor series ( 請預讀 P283~P284) Exponential function: e x. Logarithm: ln(1+x) 16   < x < .  In fact,  < x  1.

Binomial Theorem ( 請預讀 P285) Binomial Theorem: How about m  N case? Taylor’s expansion: 17

Taylor Expansion – More than one Variable ( 請預讀 P286) Proof: 18

Homework

Power series ( 請預讀 P291~P294) Power series: The key concepts The radius of convergence  uniform and absolute convergence  continuity. Uniqueness Theorem. 20

The radius of convergence ( 請預讀 P291) Based on the ratio test for absolute convergence, we get the condition of convergence of f(x) is 21

The properties of a power series ( 請預 讀 P291~P292) 22

Uniqueness Theorem ( 請預讀 P292~P293) Uniqueness Theorem: {a n } is unique. Prove: Hint: If f(x) can be represented by two different types In the convergent region, 23

Inversion of Power series ( 請預讀 P293~P294) Inverse function : f(x)=y  f -1 (y)=x. Inverse function If we know the power series of f(x), Can we find the power series of f -1 (y)? A brute-force approach: 24

Homework

Nouns 26