21-122_04.02.19 - Math 21-122 Handout 1 Power Series Definition • A power series centered at a or a power series in px ´ aq has the form 8 � cn

21-122_04.02.19 - Math 21-122 Handout 1 Power Series...

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Math 21-122 Handout 04/02/19 1 Power Series Definition: A power series centered at a , or a power series in p x ´ a q , has the form 8 ÿ n 0 c n p x ´ a q n c 0 ` c 1 p x ´ a q ` c 2 p x ´ a q 2 ` ¨ ¨ ¨ . Here c n are called coefficients. The Radius of Convergence of a power series is a positive real number R such that the series converges if | x ´ a | ă R , and diverges if | x ´ a | ą R . If | x ´ a | “ R , anything could happen – we need to examine the specific series. Rules: Given a power series f p x q “ ř 8 n 0 c n p x ´ a q n c 0 ` c 1 p x ´ a q ` c 2 p x ´ a q 2 ` ¨ ¨ ¨ , We can integrate and differentiate it by integrating and differentiating every term. More specifi- cally, f 1 p x q “ c 1 ` 2 c 2 p x ´ a q ` 3 c 3 p x ´ a q 2 ` ¨ ¨ ¨ , ż f p x q dx C ` c 0 p x ´ a q ` 1 2 c 1 p x ´ a q 2 ` ¨ ¨ ¨ . To find the Radius of Convergence R , we can use Root Test and Ratio Test. More specifically, we have lim n Ñ8 ˇ ˇ ˇ c n ` 1 c n ˇ ˇ ˇ 1 R , lim n Ñ8 n ? c n 1 R . With knowledge about geometric series, we can represent functions with power series, and represent power series with functions. See more in examples. 2 Practice Problems 1. Find the functionfpxqrepresented by the following power series:fpxq “1`2x`x2`2x3`x4`2x5` ¨ ¨ ¨ . 1
2. Find the power series representation offpxq “1p1`x2q2.3. We know that the power series representation offpxq “sinpxqisfpxq “x´x33!`x55!´x77!` ¨ ¨ ¨ “8ÿn1qnp2n`1q!x2n`1.Find the power series representation ofgpxq “cospxq.4. Find the radius of convergence of8ÿn12n?nxnand investigate the endpoints.5. Find the radius of convergence of8ÿnn23npx`2qn. 2

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