170120107066 power series.ppt
- 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)
- 3. What are Power Series?
- 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
- 6. 𝒏=𝟎 ∞ −𝟏 𝒏 𝒏 𝒙 + 𝟑 𝒏 𝟒 𝒏 Ratio Test: Convergence for 𝑳 < 𝟏 lim 𝒏→∞ −𝟏 𝒏 + 𝟏 𝒙 + 𝟑 𝟒𝒏 = 𝟏 𝟒 𝒙 + 𝟑 𝟏 𝟒 𝒙 + 𝟑 < 𝟏 𝒙 + 𝟑 < 𝟒 Radius of Convergence 𝑹 = 𝟒
- 7. Find the Radius of Convergence and the Interval of Convergence for the following power series Example: 𝒏=𝟎 ∞ 𝟐 𝒏 𝟒𝒙 − 𝟖 𝒏 𝒏 𝑐 𝑛+1 = 𝟐 𝒏+𝟏 𝟒𝒙 − 𝟖 𝒏+𝟏 𝒏 + 𝟏 Ratio Test lim 𝒏→∞ 𝟐 𝒏+𝟏 𝟒𝒙 − 𝟖 𝒏+𝟏 𝒏 + 𝟏 ÷ 𝟐 𝒏 𝟒𝒙 − 𝟖 𝒏 𝒏 lim 𝒏→∞ 𝟐 𝒏 𝟐 𝟒𝒙 − 𝟖 𝒏 𝟒𝒙 − 𝟖 𝒏 + 𝟏 ∙ 𝒏 𝟐 𝒏 𝟒𝒙 − 𝟖 𝒏 lim 𝒏→∞ 𝟐𝒏 𝟒𝒙 − 𝟖 𝒏 + 𝟏 = 𝟐 𝟒𝒙 − 𝟖
- 8. Ratio Test: Convergence for 𝑳 < 𝟏 𝟐 𝟒𝒙 − 𝟖 < 𝟏 𝟐 𝟒(𝒙 − 𝟐) < 𝟏 Radius of Convergence 𝑹 = 𝟏 𝟖 𝒏=𝟏 ∞ 𝟐 𝒏 𝟒𝒙 − 𝟖 𝒏 𝒏 lim 𝒏→∞ 𝟐𝒏 𝟒𝒙 − 𝟖 𝒏 + 𝟏 = 𝟐 𝟒𝒙 − 𝟖 𝟖 𝒙 − 𝟐 < 𝟏 𝒙 − 𝟐 < 𝟏 𝟖
- 9. 𝒏=𝟎 ∞ 𝒏! 𝟐𝒙 + 𝟏 𝒏 = ∞ > 𝟏 𝑻𝒉𝒆 𝒔𝒆𝒓𝒊𝒆𝒔 𝒘𝒊𝒍𝒍 𝒄𝒐𝒏𝒗𝒆𝒓𝒈𝒆 𝒂𝒕 𝒐𝒏𝒆 𝒑𝒐𝒊𝒏𝒕. 𝑹𝒂𝒅𝒊𝒖𝒔 𝒐𝒇 𝑪𝒐𝒏𝒗𝒆𝒓𝒈𝒆𝒏𝒄𝒆: 𝑹 = 𝟎 𝑻𝒉𝒆 𝒍𝒊𝒎𝒊𝒕 𝒊𝒔 𝒛𝒆𝒓𝒐 𝒂𝒕 𝒙 = − 𝟏 𝟐 𝑐 𝑛+1 = 𝑛 + 1 ! 2𝑥 + 1 𝑛+1Ratio Test lim 𝒏→∞ 𝑛 + 1 ! 2𝑥 + 1 𝑛+1 𝑛! 2𝑥 + 1 𝑛 lim 𝒏→∞ 𝑛! 𝑛 + 1 2𝑥 + 1 𝑛 2𝑥 + 1 𝑛! 2𝑥 + 1 𝑛 lim 𝒏→∞ 𝑛 + 1 2𝑥 + 1 Find the Radius of Convergence and the Interval of Convergence for the following power series Example: