Advanced Calculus: Power Series and Examples
Explore the concept of power series in advanced calculus through definitions and examples. Learn about convergence criteria and calculations for different series, with illustrations for better understanding.
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Advanced Calculus Advanced Calculus Second Class Second Class By By Dr. Jawad Mahmoud Dr. Jawad Mahmoud Jassim Dept. of Math. Dept. of Math. College of Education for Pure Sciences College of Education for Pure Sciences University of University of Basrah Iraq Iraq Jassim Basrah
Power Series Power Series Definition (1): A power series about ? = ? is a series of the form ????= ??+ ??? + ????+ + ???? (?) ?=? A power series about ? = ? is a series of the form ?=? ??(? ?)?= ??+ ??(? ?) + ??? ??+ + ??(? ?)?+ In which the center ? and the coefficients ??,??,??, ,??, are constants. (?)
Example (1): Take the power series ?=? ??= ? + ? + ??+ ??+ + ??+ . This is a geometric series with first term 1 and common ratio ?. It converges if |?| < ? and its sum is: ?=? ??= So, ? ? ?= ? + ? + ??+ ??+ + ??+ , ? < ? < ?. ? ? ?.
Example (2): Take the power series ? ? This is a geometric series with first term 1 and common ratio ? ? series converges for ? ? ? < ? ? Its sum is: ? ?+? ? ? ? ? ?= ? ? ? < ? < ? . ? ?? ? +? ?? ?? + ? ? ??+ ? . This ? ? ? < ? ? <? ? < ? ? < ? < ?. ? ? ? =? = ?. So ?? ? +? ? ?? ?? + ? ? ??+ , ?
Example (3): For what values of ? do the following power series converge? ( ?)? ??? ? 2: ?=? 4: ?=? ?! ?? Solution: (1): Apply the Ratio Test to the series | ??|, where ??is the nth term of the power series given in question. ??? ? ?? ? By Ratio Test, the series converges absolutely for ? < ? ? < ? < ? . ( ?)?+?? ( ?)? ???? ? ?? ?! 1: ?=? 3: ?=? ?? ? ? |??+? ??+? ? ??= ??? ?? ?+?= ? ??? ? = ??? ?+?. ?+?| = ? . ? Now, if ? = ? the series becomes ?=? This series is alternating which converges. ?.
(?)??(?)? ? If ? = ? the series becomes ?=? ?=? This series diverges because it is harmonic series. The series converges for ? < ? ?. = ?=? = ? ? ? ?. (2): Apply the Ratio Test to the series | ??|, where ??is the nth term of the power series given in question. ? |???+? ??+? By Ratio Test, the series converges absolutely for ??< ? ? < ? < ?. ??+? ?? ?? ? ??? ?= ????? ?? ? ??+?= ??. ??? ? = ??? ?
? ( ?)? ? Now, if ? = ? the series becomes ?=? This series is alternating which converges. If ? = ? the series becomes ?=? This series is alternating which converges. The series converges for ? ? ?. ?? ?. ? ( ?)? ?? ?. (3): Apply the Ratio Test to the series | ??|, where ??is the nth term of the power series given in question. ??? ? ?? (?+?)! ? . The series converges absolutely for all . ??+? ??+? ?! ??= ??? ? ?! ?+? ?!= ? ??? ? = ??? ? | ?+?| = ? ?
(4): Apply the Ratio Test to the series| ??|, where ??is the nth term of the power series given in question. ??? ? ?? ?!?? The series diverges for all values of ? except ? = ?. ?+? !??+? ??+? = ??? ? | = ? ??? ? ? + ?| = ?????? ? = ?. Theorem (1) : (The Convergence Theorem for Power Series) If the power series ?=? ????converges at ? = ? ?,then it converges absolutely for all ? with|?| < |?|. If the power series ?=? ????diverges at ? = ?,then it diverges for all ? with |?| > ? .
Note: Theorem (1) deals with the convergence of series of the form ?=? series ?=? ??(? ?)?we can replace ? ? ?? ? and apply the results to the series ?=? ????. ????. For the Corollary (1): The convergence of the series ?=? following three cases: 1: There is a positive number ? such that the series diverges for ? with |? ?| > ? but converges absolutely for ? with |? ?| < ?. The series may or may not converge at either of the end points ? = ? ? & ? = ? + ? 2: The series converges absolutely for every ? ? = . 3: The series converges at ? = ? and diverges elsewhere ? = ? . ??(? ?)?is described by one of the
Notes: 1: ? is called the radius of convergence of the power series and the interval of the radius ? centered at ? = ? is called the interval of convergence. 2: The interval of convergence may be open, closed, or half-open depending on particular series. 3: There are six different possibilities for an interval of convergence. These are ? ?,? + ? , ? ?,? + ? , ? ?,? + ? , ? ?,? + ? ,( , ) , and at ? = ?.
