University:
Amherst College
Course:
MATH 121 | Intermediate Calculus
Academic year:

2024
Views:

200

Pages:

2
Author:

vykolsinaigr

Math 121, Section(s) 01, Spring 2024 Homework #15 Due Friday, April 5th in Gradescope by 11:59 pm ET Goal: Exploring Power Series, mainly the Interval and Radius of Convergence. Also beginning to explore the relationship between Power Series and Functions. FIRST: Read through and understand the following Examples. Determine the Interval and Radius of Convergence. Justify. ∞ X an+1 (−1)n (5x − 2)n Use Ratio Test. L = lim Ex: n n→∞ (n + 5) 8 an n=1 n+1 (−1) (5x − 2)n+1 = lim n→∞ (n + 6)8n+1 n (−1) (5x − 2)n (n + 5)8n  1 |5x − 2|  n + 5 8n ·  · n+1 = n+6 8 8  = lim n→∞ (5x − 2)n+1 (5x − 2)n |5x − 2| < 1 or |5x − 2| < 8. 8 6 That is −8 < 5x − 2 < 8 =⇒ −6 < 5x < 10 =⇒ − < x < 2 5 Manually Test Endpoints: (where L = 1 and Ratio Test is Inconclusive) ∞ ∞ ∞ n X X 8 (−1)n (5(2) − 2)n X (−1)n (−1)n = •x = 2 The original series becomes = n (n + 5) 8n (n + 5) n+5 8 n=1 n=1 n=1 The Ratio Test gives convergence for x when 1 1 >0 2. lim bn = lim =0 n→∞ n→∞ n + 5 n+5 1 1 < = bn ⇒ x = 2 is Included in the Domain. 3. Terms decreasing: bn+1 = n+6 n+5 n 6 n −2 5 − ∞ (−1) ∞ X X 6 (−1)n (−8)n 5 •x = − The original series becomes = 5 (n + 5) 8n (n + 5) 8n n=1 n=1 which is Convergent by AST: 1. bn = = ∞ n X (−1)n (−1)n 8 n=1 n (n + 5) 8 ∞ ∞ ∞ :1 X X X 2n 1 1 (−1) = = ≈ the Div Harmonic p-Series p = 1. n+5 n + 5 n=1 n n=1 n=1 1 ∞ 1 X > n 1 LCT: lim n + 5 = lim = 1 which is Finite and Non-zero. Therefore, is 1 n→∞ n→∞ n+5 n + 5 n=1 n 6 also Divergent by LCT ⇒ x = − is NOT included in the Domain. 5 6 8 Finally, Interval of Convergence I = − , 2 with Radius of Convergence R = . 5 5 Ex: ∞ X n=0 x2n+1 (2n + 1)! Use Ratio Test. L = lim n→∞ an+1 an x2(n+1)+1 x2n+3 (2n + 1)! x2 (2(n + 1) + 1)! = lim = lim = lim ∞ 0 < 1 for all x := n→∞ x2n+1 (2n + 3)! n→∞ n→∞ x2n+1 (2n + 3)(2n + 2) (2n + 1)! Converges by the Ratio Test for all x and I = (−∞, ∞) with R = ∞ . Ex: ∞ X nn (x − 7)n Use Ratio Test. L = lim n=0 n→∞ an+1 an *e (n +1) (n + 1)n+1 (x − 7)n+1 n :∞ = lim n ( = lim n + 1)|x − 7| = ∞ > 1 Diverges by the Ratio n n n→∞ n n→∞ n (x − 7) Test for all x unless x − 7 = 0 or x = 7. So I = {7} with R = 0 . Homework: Determine the Interval and Radius of Convergence for each of the following Power Series. Use the Ratio Test and manually check convergence at the Endpoints for the Finite Intervals. Follow the examples above for statements/format for all three cases. ∞ X xn 1. n! n=0 2. ∞ X n=1 xn n4 · 4n 3. n! ln n (x − 6)n n=1 ∞ X (−1)n (9x − 4)n 4. n8 · 5n n=1 5. ∞ X (−1)n x2n 7. (2n)! n=0 ∞ X (−1)n (3x − 5)n 8. (n + 6)2 · 7n+1 n=1 ∞ X ∞ X n (3n)! (2x − 1) n=0 ∞ X (−1)n (6x + 1)n 6. (6n + 1) · 7n n=1 Find the Power Series Representation for the following functions and determine the Interval of Convergence. 9. f (x) = 1 1+x 10. f (x) = 5 1 − 4x 11. f (x) = 1 3−x

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