Problem 0.3 diverge: (1) m=0 1+n³ (2) Σ^=15"+3" (3) Σn=0 1-n+n² 1 In(n) Determine whether the following series converge c
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- page 6 6. (a) Find a simplified power series representation (in terms of x") for g(x) = 1 and determine its 1+3x radius of convergence. Express your answer in summation form E=o Cnx". (b) Use the power series for g(x) = to find a power series for f (x) = radius of convergence. Express your answer in either summation form E-o Cnx" or E=1 Cnx" . and determine its %3D 1+3x (1+3x)2 m%3D12(-1)"+1, n2 converges or diverges. Problem 6. Determine whether the series n3 +1 n=15. Use an appropriate test to determine the convergence or divergence of each series. Identify the test used. (a) 5 3n²+1 (b) 5(-)" 5n2+3 n=1 n=1 (c) E n2 Vn n=1 (d) Ž 2n n=1 (f) I 6"+n² 6n+n4 n=1 (e) -2n e n=0 (8) Č 3" 3+7n n=0 (h) Č Vn 9n2+4 n=1 n=2
- (6) Determine if the alternating series converges Σ(-1)^n +1. n=25. Suppose that the power series E Cnx" converges when x = 3 and diverges when x = 6. Which of the following n=0 statements is certain to be true? (a) None of these statements is certain to be true. (b) Σ Cn(-4)" is convergent. n=0 (c) Σ Cn(-3)" is convergent. n=0 (d) > Cn(-2)" is convergent. n=0 (e) 2 Cn(-6)" is divergent. n=0I just need part B. I got that the power series representation for the function is: ((-1)^n(x^6n+2))/(8^n+1)
- Consider the series (-1)n*(3n2+n+5)/(4n3+2) = 5/2 - 3/2 + 19/32 - 7/224. Does the series formed by subtracting the series 1 from the series 2n - 1 n=1 n3D1 converge? Give reasons for your answer.ncSqgHuCpmyhO2nvmCPguu9empi-50FsWnfm42xQ/formResponse Question* By shifting the summation index of the following added series +00 п(п - 1)С, х" - 4 п(п — 1)С, х"-2+3 nC,x" + C,x" = 0 n=2 we get: E[(k – 2)(k – 3)C 2C, - Co + 6C,x +£t(k + 1)(k – 1)G, +(k + 2)(k + 1)C+z]x* = 0 +(k + 2)(k + 1)Cr+2]x* = 0 O This option This option (C, - 8C;) + (4C, - 24C,)x 2C, + Co + E(1 – 2k)Cx +EI(k + 1)°C – 4(k + 2)(k+1)Gr+z]r* = 0 +(k + 2)(k + 1)Ck+2]xk = 0 O This option This option
- EXAMPLE 5 Express 16/(1 - 4x)² as a power series by differentiating the equation below. What is the radius of convergence? 4 4(1 + 4x + 16x² + 64x³ + ...) = 4 £ (4x)" n=0 (1 - 4x) SOLUTION Differentiating each side of the equation, we get 16 -= 4(4 + + 192x2 + ...) (1 - 4x)2 = 4 E n=1 If we wish, we can replace n by n + 1 and write the answer as 16 -= 4 E (1 - 4x)² n=0 According to Theorem 2, the radius of convergence of the differentiated series is the same as the radius on convergence of the original series, namely, R =Determine the sum of the series 00 n(n + 2) n=1 if possible. (If the series diverges, enter 'infinity', '-infinity' or 'dne' as appropriate.)The alternating series En=4 s+o (-1)"(1-n) Is 3n-n2 (A) (i) (ii) (iv) Converge Diverge Inconclusive