1. Differential Equations MTH 242 Lecture # 18 Dr. Manshoor Ahmed

2. Summary(Recall) Some problems related to forced damped and un-damped motion. Transient and steady state solution. Application of second order equation to electric circuits (Series circuit). Resistor, Inductor, Capacitor. Ohm’s Law, Kirchhoff’s law. Derivation of equation of motion of charge in a series. Solution of equation. Solution of some problems.

5. Convergence and Divergence If we choose a specified value of the variable x then the power series becomes an infinite series of constants (numbers). If, for the given x, the series of numbers converges (sum of terms of the power series equals a finite real number), then the series (power series) is said to be convergent at x. A power series that is not convergent is said to be a divergent series. This means that the sum of terms of a divergent power series is not equal to a finite real number.

10. Or Therefore, it follows from the Ratio Test that the power series converges absolutely for or The series diverges

11. The radius of convergence this series is 1.

13. The radius of convergence this series is 2.

15.  If a power series in x. Then the first two derivatives are  It is important to note that the first term in the first derivative and first two terms in the second derivative are zero. We omit these zero terms and write and

19. Adding Two Power Series

21. Note : The summation index is a “dummy” variable.

22. Exercises for practice Do problems 1-14 of Exercise 6.1 of your text book

23. Summary Power series. Interval convergence and radius of convergence of a power series. Ratio test for power series. Series identically zero. Analyticity of power series at a point. Arithmetic of power series. Adding two power series.