Convergence Of Power Series , Taylor And Laurent Theorems (Without Proof), Complex Variables and Numerical Methods(2141905)

1. Gandhinagar Institute
OF Technology(012)
Subject:- CVNM(2141905)
Active Learning Assignment
Topic Name:-Convergence Of Power Series , Taylor And
Laurent Theorems (Without Proof)
Branch:- Mechanical B-1
Prepared By:- Jani Parth U.(150120119051)
Guided By:-Prof. Ravikumar Panchal

2. Content
i. Introduction
ii. Power Series
iii. Convergence Of Power Series
iv. Example 1
v. Taylor’s Series And Laurent’s Series
vi. Example 2

3. i. Introduction
 A Sequence Is List (Separated By Commas) Example:-sequence:
1, 2, 3, 4, …, n, …
 A Sequence Is Denoted By 𝑍 𝑛
 A series adds the numbers in the list together. Example:-Series: 1
+ 2 + 3 + 4 + …+ n + …
 The Sum Of The Term Of Sequence Is Denoted By 𝑛=1
∞
𝑍 𝑛
 Sequence 𝑍 𝑛 Is Said To Be Convergent If Lim
𝑛→∞
𝑍 𝑛 = 𝐿
 Where L Is Finite Quantity

4. ii. Power Series
 A Power Series Is An Infinite Series Of The Form
 𝑛=0
∞
𝑎 𝑛 𝑧 − 𝑧0
𝑛
= 𝑎0 + 𝑎1(𝑧 − 𝑧0) + 𝑎2 𝑧 − 𝑧0
2
+ 𝑎 𝑛 ( 𝑧 −

5. Convergence of power series
 Radius Of Power Series :-
 Where R Is Known As Radius Of Power Series
Circle Of Power Series :-
 𝑧 − 𝑧0 = 𝑅 Is Known As Circle Of Power Series
Theorem 1:-
 A Power Series 𝑛=0
∞
𝑎 𝑛 𝑧 − 𝑧0
𝑛
And K Times Differential Series
Have The Same Radius Of Convergence

6. …
Theorem 2:-
 Let 𝑛=0
∞
𝑎 𝑛 𝑧 − 𝑧0
𝑛
Be The Power Series With Radius R
 Then, R = Lim
𝑛→∞
𝑎 𝑛
𝑎 𝑛+1
Absolute Convergence The series 𝑛=1
∞
𝑢 𝑛Is Said To Absolute
Convergence If The Series 𝑛=1
∞
𝑢 𝑛 Is Also Convergent
Uniform convergence the series 𝑛=1
∞
𝑢 𝑛(z) is said to be uniformly
convergent in region R if there exists a convergent series of
positive real constant 𝑛=1
∞
𝑚 𝑛 such that

7. iv. Example 1
 Discuss The Convergence Of 𝑛=0
∞ (2𝑛)!
(𝑛!)2 (𝑧 − 3𝑖) 𝑛
And Also Find The
Radius And Circle Of Convergence
 Solution
𝑎 𝑛 =
(2 𝑛)!
(𝑛!)2
Radius Of Convergence Is
R = Lim
𝑛→∞
𝑎 𝑛
𝑎 𝑛+1
=Lim
𝑛→∞
(2𝑛)! (𝑛+1)! 2
(𝑛!)2 2(𝑛+1) !
=Lim
𝑛→∞
(2𝑛)!(𝑛+1)2(𝑛!)2
(𝑛!)2(2𝑛+2)(2𝑛 +1)(2𝑛)!

8. …
=Lim
𝑛→∞
(𝑛+1)2
(2𝑛+2)(2𝑛 +1)
=
Lim
𝑛→∞
lim
𝑛→∞
1+
1
𝑛
2
2+
2
𝑛
2+
1
𝑛
=
1 + 0
(2 + 0)(2 + 0)
=
1
4
Type equation here.Hence Circle Convergence Is 𝑍 − 3𝑖 =
1
4
The Series Is Convergent In The Region 𝑍 − 3𝑖 <
1
4
And Is Divergent
In The Region 𝑍 − 3𝑖 >
1
4

9. v. Taylor’s and Laurent’s Series
 This Is Taylor’s Series Formula
 𝑓 𝑧 = 𝑓 𝑎 + 𝑧 − 𝑎 𝑓 𝑎 +
(𝑧−𝑎)2
2!
𝑓"(𝑎)
+…+
(𝑧−𝑎) 𝑛
𝑛!
𝑓 𝑛
𝑎 + ⋯
 𝑖𝑡 𝑖𝑠 𝑡ℎ𝑒 𝑓𝑜𝑟𝑚𝑢𝑙𝑎 𝑜𝑓 𝑙𝑎𝑢𝑟𝑒𝑛𝑡′
𝑠 𝑠𝑒𝑟𝑖𝑒𝑠 𝑒𝑥𝑝𝑎𝑛𝑠𝑖𝑜𝑛
 𝑓 𝑧 = 𝑎0 + 𝑎1 𝑧 − 𝑎 + 𝑎2 𝑧 − 𝑎 2
+ ⋯ +
𝑏1
𝑧−𝑎
+
𝑏2
(𝑧−𝑎)2 + ⋯
 F(z)= 𝑛=0
∞
𝑎 𝑛 (z−a) 𝑛
+ 𝑛=0
∞
𝑏 𝑛 /(z−a) 𝑛

10. vi. Example 2
 Expand 𝑓 𝑧 = 𝑒 𝑧
In Taylor’s Series About Z=0
 Solution
𝑓 𝑧 = 𝑒 𝑧
a=0
By Taylor’s Series
𝑓 𝑧 = 𝑓 𝑎 + 𝑓′
𝑎 𝑧 − 𝑎 +
𝑓′′ 𝑎 (𝑧−𝑎)2
2!
+
𝑓′′′ 𝑎 (𝑧−𝑎)3
3!
+ ⋯
Putting a=0
𝑓 𝑧 = 𝑓 0 + 𝑓′
0 𝑧 +
𝑓′′
0 𝑧2
2!
+
𝑓′′′
0 𝑧3
3!
+ ⋯

11. …
𝑓 𝑧 = 𝑒 𝑧
𝑓′
𝑧 = 𝑒 𝑧
𝑓′′
𝑧 = 𝑒 𝑧
𝑓′′′
𝑧 = 𝑒 𝑧
𝑓 0 = 𝑒0
=1
𝑓′
0 = 𝑒0
=1
𝑓′′
0 = 𝑒0
=1
𝑓′′′
0 = 𝑒0
=1
𝑒 𝑧
= 1 + 𝑧 +
𝑧2
2
+
𝑧3
6
+ ⋯