1. Subject : calculus(2110014)
Topic : power series and radius of
convergence.
Gandhinagar Institute of Technology(012)
Active Learning Assignment
Guided By:prof. keyuri mam
Prepared By:Harsh kothari
Branch:CE-B
Enrolment no.:170120107066
2. Previous series have consisted of constants.
𝒏=𝟏
∞
𝒂 𝒏
Another type of series will include the variable x.
𝒏=𝟏
∞
𝟏
𝒏
𝒏=𝟏
∞
𝟏𝟎 𝒏
𝒏 + 𝟏 !
𝒏=𝟏
∞
−𝟏 𝒏
𝒍𝒏(𝒏)
𝒏 − 𝒍𝒏(𝒏)
4. There are only three ways for a power series to converge.
1) The series only converges at 𝒙 = 𝒂.
2) The series converges for all x values.
3) The series converges for some interval of x.
Interval of Convergence: The interval of x values where the series converges.
Radius of Convergence (R): Half the length of the interval of convergence.
Definitions
(𝑹 = 𝟎)
(𝑹 = ∞)
(𝒂 − 𝑹 < 𝒙 < 𝒂 + 𝑹)
The end values of the interval must be tested for convergence.
𝒙 − 𝒂 < 𝑹
The use of the Ratio test is recommended when finding the radius of
convergence and the interval of convergence.
5. Find the Radius of Convergence and the Interval of Convergence
for the following power series
Example:
𝒏=𝟎
∞
−𝟏 𝒏 𝒏 𝒙 + 𝟑 𝒏
𝟒 𝒏 𝑐 𝑛+1 =
−𝟏 𝒏+𝟏
𝒏 + 𝟏 𝒙 + 𝟑 𝒏+𝟏
𝟒 𝒏+𝟏
Ratio Test
lim
𝒏→∞
−𝟏 𝒏+𝟏
𝒏 + 𝟏 𝒙 + 𝟑 𝒏+𝟏
𝟒 𝒏+𝟏
÷
−𝟏 𝒏
𝒏 𝒙 + 𝟑 𝒏
𝟒 𝒏
lim
𝒏→∞
−𝟏 𝒏 −𝟏 𝒏 + 𝟏 𝒙 + 𝟑 𝒏 𝒙 + 𝟑
𝟒 𝒏 𝟒
∙
𝟒 𝒏
−𝟏 𝒏 𝒏 𝒙 + 𝟑 𝒏
lim
𝒏→∞
−𝟏 𝒏 + 𝟏 𝒙 + 𝟑
𝟒𝒏
=
𝟏
𝟒
𝒙 + 𝟑