Power Series

So, start asking yourself this question: Can x be 0, 1 / 2, – 1 / 2, 3 / 4, – 3 / 4, 1, –1, 3 / 2, – 3 / 2, 2, –2, and so on? Power SeriesIntroduction A power series is a series of the form c n x n = c 0 + c 1 x + c 2 x 2 + c 3 x n = 0 Example 1: Say c n = 1 for all n, we have x n as our first power series. n = 0 Well, what we really want to know is: for what values of x will x n converge? n = 0 Yes, when x is any of these numbers, x n becomes convergent geometric series since |r| < 1 n = 0 No, when x is any of these numbers, x n becomes divergent geometric series since |r| 1 n = 0 Conclusion: x n converges when x is any number between –1 and 1, i.e., –1 < x < 1, or the open interval: (–1, 1). n = 0 So?

Power SeriesInterval and Radius of Convergence Recall our power series x n converges for x (–1, 1). Therefore, we say (–1, 1) is the interval of convergence (IOC) and we define the radius of convergence (ROC) as half of the length of IOC. Since the length of the interval (i.e., the distance from –1 to 1) is 2, therefore, the ROC is 1. n = 0 –101 Interval of Convergence = (–1, 1) Radius of Convergence = 1 Again, ask yourself this question: Can x be 0, 1 / 2, – 1 / 2, 3 / 4, – 3 / 4, 1, –1, 3 / 2, – 3 / 2, 2, –2, and so on? Yes, if x is 0, it will make every term 0 Example 2: n!x n n = 0 No, if x is any of these numbers, n!x n will diverge n = 0 IOC = {0} –101 ROC = 0

Power SeriesThe Three Types of IOCs Once more, ask yourself this question: Can x be 0, 1 / 2, – 1 / 2, 3 / 4, – 3 / 4, 1, –1, 3 / 2, – 3 / 2, 2, –2, and so on? –202–11 Interval of Convergence = (–, ) Radius of Convergence = Yes, if x is any of these numbers, x n /n! will converge n = 0 The Three Types of IOCs: For any given power series c n x n, there are only three possibilities: (1) The series converges on an interval with a finite length ( Example 1 ) (2) The series converges at only one number ( Example 2 ) (3) The series converges for all real numbers ( Example 3 ) n = 0 Example 3: x n /n! n = 0

Power SeriesMore on IOCs and ROCs IOCROC [1, 4)½(3) = 1.5 (–2, 2]½(4) = 2 Examples: 1) 2) 3) –213–120 –213–120 {½}0

Power SeriesHow to Guess the IOC? One number, in particular, will obviously make the series converge. What is this number? [Hint: this number will make every term = 0.] Example 1: x = 3 Of course, x may be other numbers too. If so, on the number line, we begin at 3 and move to the right and left to obtain our interval of convergence Therefore, the IOC is [2, 4), and ROC = 1. If x = 4, then the series becomes which is a divergent series If x = 2, then the series becomes which is a convergent series

Power SeriesHow to Guess the IOC? (contd) What is the number that makes every term = 0? x = 1 Again, on the number line, we begin at 1 and move to the right and left to obtain our interval of convergence Therefore, the IOC is [0, 2], and ROC = 1. If x = 2, then the series becomes which is a convergent series If x = 0, then the series becomes which is a convergent series Example 2:

Power SeriesHow to Find the IOC in General? Answer: Kind of like doing the Ratio Test Example 1: |x – 3| < 1 –1 < x – 3 < 1 2 < x < and Step 1: Find : Step 2: Find : Step 3: Set above limit < 1 and solve for x: Step 4: Check whether endpoints work or not: x – 3 If x = 2, the series becomes which is a convergent series, so 2 is included. If x = 4, the series becomes which is a divergent series, so 4 should be excluded. IOC is [2, 4), and ROC = 1.

Power SeriesHow to Find the IOC in General? (contd) Recap: 2) We then check the endpoints (to see whether they will make the power series converges or not) by substituting each endpoint into the series. ( Step 4 ) 1) We find and set it < 1, and solve for x. ( Steps 1–3 ) Example 2:Example 3:Example 4: