**Chapter 7** **Infinite Series**

**(無窮級數)**

**Hung-Yuan Fan (范洪源)**

**Department of Mathematics,**
**National Taiwan Normal University, Taiwan**

**Spring 2019**

## 本章預定授課範圍

**7.1 Sequences**

**7.2 Series and Convergence**

**7.3 The Integral and Comparisons Tests** **7.4 Other Convergence Tests**

**7.5 Taylor Polynomials and Approximations** **7.6 Power Series**

**7.7 Representation of Functions by Power Series**

**7.8 Taylor and Maclaurin Series**

**Section 7.1** **Sequences**

**(序列)**

## Sequences (序列)

An infinite sequence of real numbers is denoted by
*{a**n**} = {a**n**}*^{∞}*n=1*=*{a*1*, a*2*,· · · , a**n**,· · · },*

*where a*_{n}*is the nth term (第 n 項) of the sequence for n∈ N.*

**Def. (序列的收斂性)**

A sequence*{a**n**} converges to a limit L, denoted by*

*n*lim*→∞**a*_{n}*= L,*
if*∀ ε > 0, ∃ M > 0 s.t.*

*n > M =⇒ |a**n**− L| < ε.*

Otherwise, we say that*{a**n**} diverges if the limit does not exist.*

## 示意圖 (承上頁)

**Thm 7.1 (函數在** **∞ 處的極限 ⇒ 序列的收斂性)**

**∞ 處的極限 ⇒ 序列的收斂性)**

*Let f be a real-valued function having the limit*

*x*lim*→∞**f(x) = L.*

If*a*_{n}*= f(n)* *∀ n ∈ N*, then

*n*lim*→∞**a*_{n}*= L.*

**pf: Let ε > 0 be given arbitrarily. Since lim**

**pf: Let ε > 0 be given arbitrarily. Since lim**

*x**→∞**f(x) = L,∃ M > 0 s.t.*

*x > M =⇒ |f(x) − L| < ε.*

With*a*_{n}*= f(n) for all n∈ N*, if*n > M, then*

*|a**n**− L| = |f(n) − L| < ε.*

Thus, it follows from the Def. that

*n*lim*→∞**a** _{n}*= lim

*n**→∞**f(n) = L.*

**Example (Thm 7.1 的反敘述不成立)**

*Consider f(x) = sin(πx) for all x > 0 and let a*_{n}*= f(n) for n∈ N.*

Then

*n*lim*→∞**a** _{n}*= lim

*n**→∞**f(n) = lim*

*n**→∞**sin(nπ) = lim*

*n**→∞**0 = 0,*
but we know that

*x*lim*→∞**f(x) = lim*

*x**→∞**sin(πx)* does not exist!

**Thm 7.2 (Limit Laws for Sequences)**

Suppose that lim
*n**→∞**a*_{n}*= L and lim*

*n**→∞**b*_{n}*= K. Then*

**1** lim

*n**→∞**(a**n**± b**n**) = L± K.*

**2** lim

*n**→∞**(c· a**n**) = c· L for all c ∈ R.*

**3** lim

*n**→∞**(a**n**· b**n**) = L· K.*

**4** lim

*n**→∞*

*a*_{n}*b** _{n}* =

*L*

*K* if*b*_{n}*̸= 0 ∀ n ∈ N* and*K̸= 0*.

**Thm 7.3 (Squeeze Theorem for Sequences)**

If*∃ M > 0 s.t.a*

_{n}*≤ c*

*n*

*≤ b*

*n*

*∀ n > M*, and

*n*lim*→∞**a*_{n}*= L = lim*

*n**→∞**b*_{n}*,*

then the sequence*{c**n**} converges to the same limit L, i.e.,*

*n*lim*→∞**c*_{n}*= L.*

**How to find the bounds for** *{c*

*n*

**}?**

**}?**

Since*n!≥ 2** ^{n}* for all

*n≥ 4*, it follows that

*|c**n**| =*(*−1)*^{n}*n!*

= 1
*n!* *≤* 1

2^{n}*,*
and hence we immediately obtain

*−1*

2^{n}*≤ c**n**≤* 1
2* ^{n}*
for all

*n≥ 4*.

**Thm 7.4 (Absolute Value Thoerem)**

Let*{a*

*n*

*} be a sequence of real numbers. Then*

*n*lim*→∞**|a**n**| = 0 ⇐⇒ lim*_{n}

*→∞**a*_{n}*= 0.*

## Monotonic and Bounded Sequences (1/2)

**Def. (單調序列的定義)**

A sequence*{a**n**} is said to be monotonic (單調的) if its terms are*
nondecreasing (遞增的)

*a*_{n}*≤ a**n+1* *∀ n ∈ N,*
or its terms arenonincreasing (遞減的)

*a*_{n}*≥ a**n+1* *∀ n ∈ N.*

## Monotonic and Bounded Sequences (2/2)

**Def. (有界序列的定義)**

(1) The sequence *{a**n**} is*bounded above (於上有界) if*∃ M ∈ R*
s.t. *a*_{n}*≤ M ∀ n ∈ N*.

(2) The sequence *{a**n**} is*bounded below (於下有界) if*∃ N ∈ R*
s.t. *a*_{n}*≥ N ∀ n ∈ N*.

(3) The sequence*{a**n**} is* bounded (有界的)if it isbounded above
and bounded below, i.e. *∃ M > 0 s.t. |a**n**| ≤ M ∀ n ∈ N.*

**Thm 7.5 (Bounded Monotonic Sequences)**

If the sequence*{a**n**} is*bounded and monotonic, then it converges,
i.e.,*∃! L ∈ R s.t. lim*_{n}

*→∞**a*_{n}*= L.*

**Example 9 (Thm 7.5 的例子)**

(a) The sequence *{a**n**} = {1/n} is bounded and nonincreasing,*
since

*|a**n**| ≤ 1 and a**n*= 1
*n* *≥* 1

*n + 1* =*a*_{n+1}*∀ n ∈ N.*
So, it must converge by Thm 7.5 with lim

*n**→∞**a** _{n}*= 0.

**Thm 7.5 (Bounded Monotonic Sequences)**

If the sequence*{a**n**} is*bounded and monotonic, then it converges,
i.e.,*∃! L ∈ R s.t. lim*_{n}

*→∞**a*_{n}*= L.*

**Example 9 (Thm 7.5 的例子)**

(a) The sequence *{a**n**} = {1/n} is bounded and nonincreasing,*
since

*|a**n**| ≤ 1 and a**n*= 1
*n* *≥* 1

*n + 1* =*a*_{n+1}*∀ n ∈ N.*

So, it must converge by Thm 7.5 with lim

*n**→∞**a** _{n}*= 0.

**Section 7.2**

**Series and Convergence**

**(級數與收斂)**

**Def. (Partial Sums of a Series)**

(a) An infinite series (無窮級數) of real numbers is of the form
X*a** _{n}*=

X*∞*
*n=1*

*a*_{n}*= a*1*+ a*2*+ a*3+*· · · .*

(b) *For each n∈ N, the nth partial sum (第 n 個部分和) of*P
*a** _{n}*
is defined by

*S** _{n}*=
X

*n*

*i=1*

*a*_{i}*= a*_{1}*+ a*_{2}+*· · · + a**n**.*

**Def. (Convergence of a Series)**

(a) We say thatP
*a** _{n}* converges if the sequence

*{S*

*n*

*} converges*with lim

*n**→∞**S*_{n}*= S. In this case, S is called the sum of the*
*series and write S =*P

*a** _{n}*.
(b) We say thatP

*a** _{n}* diverges if the sequence

*{S*

*n*

*} diverges.*

## Two Questions for Series

**Two Questions**

**1** Does a given series converge or diverge?

**2** *What is the sum of a convergent series?*

**Note: these questions are not always easy to answer,**

especially
the second one. (通常需要藉助數值方法求得近似和!)
## Geometric Series (幾何級數)

*For a̸= 0 and ratio (公比) r ̸= 0, the geometric series is given by*
X*∞*

*n=0*

*ar*^{n}*= a + ar + ar*^{2}*+ ar*^{3}+*· · · .*

**Thm 7.6 (幾何級數的收斂性)**

X*∞*
*n=0*

*ar** ^{n}*=

*a*

1*− r* if *|r| < 1,*
diverges if*|r| ≥ 1.*

**Example 3 (Thm 7.6 的例子)**

(a)
X*∞*
*n=0*

3
2* ^{n}* =

X*∞*
*n=0*

3

1 2

*n*

= 3

1*− (1/2)* = 6 by Thm 7.6.

(b)
X*∞*
*n=0*

3 2

*n*

diverges because the ratio*|r| = |3/2| = 3/2 ≥ 1.*

**Thm 7.7 (Properties of Infinite Series)**

Suppose thatP
*a*_{n}*= A and* P

*b*_{n}*= B are convergent series.*

**1** P

*(c· a**n**) = c·* P
*a*_{n}

*= c· A for all c ∈ R.*

**2** P

*(a**n**± b**n*) = P
*a*_{n}

*±* P
*b*_{n}

*= A± B.*

**3** In general, we know that
X*(a**n**b** _{n}*)

*̸=*X

*a** _{n}* X

*b*

_{n}*,* *X a*_{n}*b*_{n}

*̸=*

P*a** _{n}*
P

*b*

_{n}*.*

**Thm 7.8**

IfP
*a** _{n}* converges, then lim

*n**→∞**a** _{n}*= 0.

*(收斂級數第 n 項所形成的序列必定趨近 0!)*

**pf: If the series**

P
*a*_{n}*= S converges, then we know that*

*n*lim*→∞**S*_{n}*= S = lim*

*n**→∞**S*_{n}_{−1}*,*
*where S** _{n}*=

P*n*
*i=1*

*a*_{i}*is the nth partial sum of*P

*a** _{n}*. So, we
immediately obtain

*n*lim*→∞**a** _{n}*= lim

*n**→∞**(S*_{n}*− S**n**−1**) = S− S= 0.*

**Thm 7.9 (The nth Term Test; 第 n 項測試法)**

If lim
**Thm 7.9 (The nth Term Test; 第 n 項測試法)**

*n**→∞**a*_{n}*̸= 0*, the series P

*a** _{n}* diverges.

*(若第 n 項不趨近到 0, 則級數必定發散!)*

**Note: 此定理為 Thm 7.8 的反敘述，常用於判斷級數的發散性。**

## Divergence of Harmonic Series (示意圖)

**Section 7.3**

**The Integral and Comparisons Tests**

**(積分與比較測試法)**

**Thm 7.10 (The Integral Test; 積分測試法)**

*If f is*positive, conti. and*↘ on [1, ∞)*, and*a*_{n}*= f(n)* *∀ n ∈ N*,
then

X*∞*
*n=1*

*a** _{n}* and
Z

_{∞}1

*f(x) dx*both converge or both diverge.

**Note: 上述定理只說明級數與瑕積分是同收同發，但並沒有說明**

兩者相等喔!
**Def. (p-級數的定義)**

(a)
**Def. (p-級數的定義)**

X*∞*
*n=1*

1

*n*^{p}*is called a p-series (p-級數) with* *p > 0.*

(b) *If p = 1, then*
X*∞*
*n=1*

1

*n* is called a harmonic series (調和級數).

**Thm 7.11 (p-級數的收斂與發散)**

**Thm 7.11 (p-級數的收斂與發散)**

*The p-series*

X*∞*
*n=1*

1

*n** ^{p}* converges for

*p > 1, and diverges for*

*0 < p≤ 1*.

**Note: 此定理只探討 p-級數的收斂性，但無法由此求出該級數**

的和!
**Note: 此定理只探討 p-級數的收斂性，但無法由此求出該級數**

**計算 p-級數的和 (檔名: p_series.m)**

p = 2.7; data = [];
**計算 p-級數的和 (檔名: p_series.m)**

for k = 0:8 N = 10^k;

n = 1:N;

S_N = sum(1./(n.^p));

data = [data;N S_N];

end

semilogx(data(:,1),data(:,2),'bo-');

title('Partial sums of a p-series with p = 2.7');

xlabel('*\bf n');*

ylabel('*\bf S_n');*

P*∞*

## 程式執行結果 (承上例)

下列圖形顯示 P^{∞}

*n=1*

*n*1^{p}*的收斂與發散，其中 p = 2.7 和 p = 0.7:*

**Thm 7.12 (Direct Comparison Test; 直接比較法)**

If*0 < a*

*n*

*≤ b*

*n*

*for all n∈ N, then*

**1** P

*b** _{n}*converges =

*⇒*P

*a** _{n}*converges, and

**2** P

*a** _{n}* diverges =

*⇒*P

*b** _{n}* diverges.

**口訣:**

(1) 大的級數收斂保證小的級數也收斂!

(2) 小的級數發散保證大的級數也發散!

**Thm 7.13 (Limit Comparison Test; 極限比較法)**

Let*a*

_{n}*, b*

*n*

*> 0*

*∀ n ∈ N*with lim

*n**→∞*

*a*_{n}*b*_{n}*= L.*

**1** If*0 < L <∞*, then P

*a** _{n}*and P

*b** _{n}* both converge or both
diverge.

**2** If*L = 0, then* P

*b** _{n}* converges =

*⇒*P

*a** _{n}* converges, and
P

*a*

*diverges =*

_{n}*⇒*P

*b** _{n}* diverges.

**3** If*L =∞*, then P

*a** _{n}*converges =

*⇒*P

*b** _{n}* converges, and
P

*b*

*diverges =*

_{n}*⇒*P

*a** _{n}* diverges.

**Section 7.4**

**Other Convergence Tests**

**(其他收斂測試法)**

**Alternating Series (交錯級數)**

An alternating series is of the form
*S =*
X*∞*
*n=1*

(*−1)*^{n+1}*a** _{n}*=
X

*∞*

*n=1*

(*−1)*^{n}^{−1}*a*_{n}*= a*_{1}*− a*2*+ a*_{3}*− a*4+*· · · ,*

*where the nth terma*_{n}*> 0 for all n∈ N*.

**Questions**

*Does the alternating series always converge?*

*How to estimate the sum S of a convergent alternating series?*

**Thm 7.14 (Alternating Series Test; 交錯級數測試法)**

If the sequence ofpositive terms*{a*

*n*

*} satisfies*

**1** *∃ M > 0 s.t. a**n**≥ a**n+1* *for all n > M, and*

**2** lim

*n**→∞**a** _{n}*= 0,

*then the alternating series S =*P

(−1)^{n+1}*a** _{n}* converges with the
property

*|S**n**− S| ≤ a**n+1* *∀ n ∈ N.*

## Absolute and Conditional Convergence

**Def. (Types of Convergence for a Series)**

(1) P
*a** _{n}* is absolutely convergent (絕對收斂) if P

*|a**n**| converges.*

(2) P

*a** _{n}* is conditionally convergent (條件收斂) if P

*a*

*converges, but P*

_{n}*|a**n**| diverges.*

**Thm 7.16 (絕對收斂保證原級數收斂)**

P*|a*

*n*

*| converges =⇒*P

*a** _{n}*converges.

## Absolute and Conditional Convergence

**Def. (Types of Convergence for a Series)**

(1) P
*a** _{n}* is absolutely convergent (絕對收斂) if P

*|a**n**| converges.*

(2) P

*a** _{n}* is conditionally convergent (條件收斂) if P

*a*

*converges, but P*

_{n}*|a**n**| diverges.*

**Thm 7.16 (絕對收斂保證原級數收斂)**

P*|a*

*n*

*| converges =⇒*P

*a** _{n}*converges.

**Example (Thm 7.16 的反例)**

The series P

^{∞}*n=1*
(−1)^{n}

*n* converges by the Alternating Series Test,
but P^{∞}

*n=1**|a**n**| =* P^{∞}

*n=1*
1

*n* is a *divergent p-series with p = 1!*

In fact, any conditionally convergent series gives a counterexample of Thm 7.16.

**Thm 7.17 (The Ratio Test; 比值法)**

Let*a*

_{n}*̸= 0 ∀ n ∈ N*with

*ρ = lim*

*n**→∞*

*a*_{n+1}*a*_{n}

.

**1** *ρ < 1* =*⇒*P

*a** _{n}* converges absolutely.

**2** *ρ > 1* or *ρ =∞* =*⇒*P

*a** _{n}* diverges.

**3** *ρ = 1* =*⇒ the test is inconclusive.*

**Thm 7.18 (The Root Test; 根式法)**

Let*a*

_{n}*̸= 0 ∀ n ∈ N*with

*ρ = lim*

*n**→∞*

p*n*

*|a**n**| = lim*_{n}

*→∞*

*|a**n**|*1*n*
.

**1** *ρ < 1* =*⇒*P

*a** _{n}* converges absolutely.

**2** *ρ > 1* or *ρ =∞* =*⇒*P

*a** _{n}* diverges.

**3** *ρ = 1* =*⇒ the test is inconclusive.*

**Section 7.5**

**Taylor Polynomials and**

**Approximations**

**(泰勒多項式與近似)**

**Def. (Taylor and Maclaurin Polynomials)**

*Suppose that f has n derivatives at c∈ dom(f).*

(1) A polynomial of the form

*P*_{n}*(x) =*
X*n*
*k=0*

*f*^{(k)}*(c)*

*k!* *(x− c)*^{k}

*= f(c) + f*^{′}*(c)(x− c) + · · · +* *f*^{(n)}*(c)*

*n!* *(x− c)*^{n}*is called the nth Taylor poly. (n 階泰勒多項式) for f at c.*

(2) If*c = 0, then P*_{n}*(x) =*
X*n*
*k=0*

*f** ^{(k)}*(0)

*k!* *x*^{k}*is called the nth*
*Maclaurin poly. (n 階馬克勞林多項式) for f.*

*多項式 P*

_{1}

*(x) 和 P*

_{2}

*(x) 的示意圖 (承上例)*

*多項式 P*

_{3}

*(x) 和 P*

_{4}

*(x) 的示意圖 (承上例)*

*多項式 P*

_{6}

*(x) 的示意圖 (承上例)*

**Def. (The Remainder of P**

**Def. (The Remainder of P**

_{n}

**(x))***Let f have (n + 1) derivatives on an interval I containing c.*

(1) *R*_{n}*(x)≡ f(x) − P**n**(x) is called the remainder (剩餘項)*
*associated with P*_{n}*(x).*

(2) *|R**n**(x)| = |f(x) − P**n**(x)| is the error associated with P**n**(x).*

**Thm 7.19 (Taylor’s Theorem; 泰勒定理)**

*If f has (n + 1) derivatives on an interval I containing c, then*

*∀ x ∈ I,∃ z between x and c* s.t.

*f(x) =*
X*n*
*k=0*

*f*^{(k)}*(c)*

*k!* *(x− c)*^{k}*+ R**n**(x) = P**n**(x) + R**n**(x),*
where the Lagrange form of the remainder is given by

*R*_{n}*(x) =* *f** ^{(n+1)}*(z)

*(n + 1)!(x− c)*^{n+1}*.*

**Section 7.6** **Power Series**

**(冪級數)**

**Def. (以 c 點為中心的冪級數)**

An infinite series of the form
**Def. (以 c 點為中心的冪級數)**

X*∞*
*n=0*

*a*_{n}*(x− c)*^{n}*= a*0*+ a*1*(x− c) + a*2*(x− c)*^{2}+*· · ·*

*is called a power series (冪級數) centered at c∈ R.*

## 冪級數的收斂行為

**Three Types of Convergence for Power Series**

For a power series P

^{∞}*n=0*

*a*_{n}*(x− c)** ^{n}*, you will see that

**1** *it converges only at x = c or*

**2** it converges only for*|x − c| < R with R > 0 or*

**3** *it converges for all x∈ R.*

## Radius and Interval of Convergence

Denote

*R = Radius of Convergence (收斂半徑),*
*I = Interval of Convergence (收斂區間).*

**Type I of Convergence**

X*∞*
*n=0*

*a*_{n}*(x− c)*^{n}*converges only at x = c.*

=*⇒R = 0 and I = {c}.*

**Type II of Convergence**

X*∞*
*n=0*

*a*_{n}*(x− c)** ^{n}*converges absolutely for

*|x − c| < R, and*diverges for

*|x − c| > R.*

=*⇒R > 0 and I = (c − R, c + R).*

**Type III of Convergence**

X*∞*
*n=0*

*a*_{n}*(x− c)*^{n}*converges for all x∈ R.*

=*⇒R = ∞ and I = (−∞, ∞).*

**Example 4 (Type III 的例子)**

Find the radius of convergence for the power series
X*∞*

*n=0*

(*−1)*^{n}

*(2n + 1)!x*^{2n+1}*.*

**Note**

In Section 7.8, we will further show that
*sin x =*

X*∞*
*n=0*

(*−1)*^{n}

*(2n + 1)!x*^{2n+1}*= x−x*^{3}
3! +*x*^{5}

5! *−* *x*^{7}

7! +*· · · ∀ x ∈ R.*

**Endpoint Convergence (端點收斂性)**

*Besides the interval of convergence I = (c− R, c + R) in Type II,*
we also have

*I = [c− R, c + R), (c − R, c + R] or [c − R, c + R]*

for different endpoint convergence of a power series.

**Example 5 (端點收斂的例子)**

Find the interval of convergence of the power series
X*∞* *x*^{n}

*= x +x*^{2}
+*x*^{3}

+ *x*^{4}

+*· · · .*

## Differentiation and Integration of Power Series

For the cases of Type II or Type III, we consider a real-valued
*function f defined by*

*f(x) =*
X*∞*
*n=0*

*a*_{n}*(x− c)*^{n}*∀ x ∈ I,*

*where I = (c− R, c + R) with R > 0, or I = (−∞, ∞).*

**Two Questions**

*Is f differentiable and integrable on the open interval I?*

*If yes, what are f* ^{′}*(x) and*
Z

*f(x) dx?*

**Thm 7.21 (冪級數的微分與積分公式)**

*Let f(x) =*

X*∞*
*n=0*

*a*_{n}*(x− c)*^{n}*be well-defied on I = (c− R, c + R).*

**1** Term-by-term differentiation (逐項微分):

*f*^{′}*(x) =*
X*∞*
*n=0*

*d*

*dx[a*_{n}*(x− c)** ^{n}*] =
X

*∞*

*n=1*

*na*_{n}*(x− c)*^{n}^{−1}*∀ x ∈ I.*

**2** Term-by-term integration (逐項積分):

Z

*f(x) dx =*
X*∞*
*n=0*

h Z *a*_{n}*(x− c)*^{n}*dx*i

=*C*+
X*∞*
*n=0*

*a*_{n}

*n + 1(x− c)*^{n+1}*∀ x ∈ I,*

**Remarks**

(1) *The radii of convergence of f* ^{′}*(x) and*
Z

*f(x) dx are the same*
*as that of f(x) =*P

*a*_{n}*(x− c)** ^{n}*.

(2) *But, their intervals of convergence may differ from f.*

*(微分和積分後的收斂半徑與 f 相同，但收斂區間略有不同!)*

**Section 7.7**

**Representation of Functions by Power** **Series**

**(以冪級數作為函數的表示式)**

**Geometric Power Series (幾何冪級數)**

*The function f(x) =*1

1*− x* *is well-defined for x̸= 1.*

*f has a geometric power series centered at x = 0, i.e.,*

*f(x) =* 1
1*− x* =

X*∞*
*n=0*

*x*^{n}*= 1 + x + x*^{2}*+ x*^{3}+*· · ·*
for *|x| < 1*.

## 示意圖 (承上頁)

**Operations with Power Series**

*Let f(x) =*

X*∞*
*n=0*

*a*_{n}*x*^{n}*and g(x) =*
X*∞*
*n=0*

*b*_{n}*x** ^{n}* be well-defined.

**1** *For any k∈ R, f(kx) =*X^{∞}

*n=0*

*a** _{n}*(kx)

*= X*

^{n}*∞*

*n=0*

(a_{n}*k*^{n}*)x** ^{n}*.

**2** *For any N∈ N, f(x** ^{N}*) =
X

*∞*

*n=0*

*a*_{n}*(x** ^{N}*)

*= X*

^{n}*∞*

*n=0*

*a*_{n}*x** ^{nN}*.

**3** *f(x)± g(x) =*X^{∞}

*n=0*

*a*_{n}*x*^{n}

*±*X^{∞}

*n=0*

*b*_{n}*x*^{n}

=
X*∞*
*n=0*

*(a**n**± b**n**)x** ^{n}*.

**Section 7.8**

**Taylor and Maclaurin Series**

**(泰勒級數與馬克勞林級數)**

**Thm 7.22 (The Form of a Convergent Power Series)**

*If f is a real-valued function defined on I = (c− R, c + R) by*

*f(x) =*
X*∞*
*n=0*

*a*_{n}*(x− c)*^{n}*∀ x ∈ I,*

*then f has derivatives of all orders on I, and moreover, we have*
*a** _{n}*=

*f*

^{(n)}*(c)*

*n!* *∀ n ∈ N,*
*with 0! = 1 and f*^{(0)} *= f.*

**Def. (泰勒級數的定義)**

*Suppose that f has derivatives of all orders at c.*

(1) A power series of the form
X*∞*

*n=0*

*f*^{(n)}*(c)*

*n!* *(x− c)*^{n}*= f(c) + f*^{′}*(c)(x− c) +f* ^{′′}*(c)*

2! *(x− c)*^{2}+*· · ·*
*is called the Taylor series (泰勒級數) for f at c.*

(2) If*c = 0, a power series of the form*
X*∞*
*n=0*

*f** ^{(n)}*(0)

*n!* *x** ^{n}*is called the

*Maclaurin series (馬克勞林級數) for f.*

## Convergence of Taylor Series

*Suppose that f has derivatives of all orders on an open interval I*
*containing c. It follows from Taylor’s Thm (Thm 7.19) that*

*∀ x ∈ I,∃ z between x and c* s.t.

*f(x) =*
X*n*
*k=0*

*f*^{(k)}*(c)*

*k!* *(x− c)*^{k}*+ R**n**(x) = P**n**(x) + R**n**(x),*
where the Lagrange form of the remainder is given by

*R*_{n}*(x) =* *f** ^{(n+1)}*(z)

*(n + 1)!(x− c)*^{n+1}*.*

**Question**

*When does the Taylor series for f at c always converge to f on the*
*open interval I?*

**Thm 7.23 (泰勒級數的收斂性)**

*f(x) =*

X*∞*
*n=0*

*f*^{(n)}*(c)*

*n!* *(x− c)*^{n}*conv. on I⇐⇒* lim

*n**→∞**R*_{n}*(x) = 0* *∀ x ∈ I*.

**Question**

*When does the Taylor series for f at c always converge to f on the*
*open interval I?*

**Thm 7.23 (泰勒級數的收斂性)**

*f(x) =*

X*∞*
*n=0*

*f*^{(n)}*(c)*

*n!* *(x− c)*^{n}*conv. on I⇐⇒* lim

*n**→∞**R*_{n}*(x) = 0* *∀ x ∈ I*.

**Remark**

In order to prove that

*n*lim*→∞**R*_{n}*(x) = lim*

*n**→∞*

*f*^{(n+1)}*(z)*

*(n + 1)!(x− c)** ^{n+1}*= 0

*∀ x ∈ I,*we often use the Squeeze Theorem and the following fact

*n*lim*→∞*

*|x − c|*^{n+1}

*(n + 1)!* = 0 *∀ x ∈ R.*

**Useful Taylor or Maclaurin Series (1/2)**

(1) 1
1*− x* =
X*∞*
*n=0*

*x** ^{n}* for

*−1 < x < 1*.

(2) *ln x =*
X*∞*
*n=1*

(*−1)*^{n}^{−1}

*n* *(x− 1)** ^{n}*for

*0 < x≤ 2*.

(3) *e** ^{x}*=
X

*∞*

*n=0*

*x*^{n}

*n!* for *−∞ < x < ∞*.

(4) *sin x =*

X*∞* (*−1)*^{n}*x** ^{2n+1}* for

*−∞ < x < ∞*.

**Useful Taylor or Maclaurin Series (2/2)**

(5) *cos x =*

X*∞*
*n=0*

(*−1)*^{n}

*(2n)!x** ^{2n}* for

*−∞ < x < ∞*.

(6) sin^{−1}*x =*
X*∞*
*n=0*

*(2n)!*

(2^{n}*n!)*^{2}*(2n + 1)x** ^{2n+1}* for

*−1 ≤ x ≤ 1*.

(7) tan^{−1}*x =*
X*∞*
*n=0*

(*−1)*^{n}

*2n + 1x** ^{2n+1}* for

*−1 ≤ x ≤ 1*. (8) Binomial Series (二項級數) with

*k∈ R*:

*(1+x)** ^{k}* = 1+

X*∞*
*n=1*

*k(k− 1) · · · (k− n + 1)*

*n!* *x** ^{n}* for

*−1 < x < 1.*