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Chem Differential Eq HW Solutions Fall 2011 188

# Chem Differential Eq HW Solutions Fall 2011 188 - A188...

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Unformatted text preview: A188 Appendix A Ordinary Diﬀerential Equations: Review of Concepts and Methods 9. Using the ratio test, we have that the series ∞ [10(x + 1)]2m (m!)2 m=1 converges whenever the following limit is < 1: [10(x + 1)]2m (m!)2 (m!)2 ((m + 1)!)2 (m!)2 (m + 1)2 (m!)2 = 102 |x + 1|2 lim m→∞ [10(x + 1)]2(m+1) ((m + 1)!)2 = 102 |x + 1|2 lim = 102 |x + 1|2 lim lim 1 = 0. (m + 1)2 m→∞ m→∞ m→∞ Thus the series converges for all x, R = ∞. 13. We use the geometric series. For |x| < 1, 3−x 1+x 1+x 4 + 1+x 1+x = − = −1 + = −1 + 4 4 1 − (−x) ∞ (−1)nxn . n=0 17. Use the Taylor series ∞ ex = xn n! n=0 Then − ∞ < x < ∞. ∞ 2 eu = u 2n n! n=0 − ∞ < u < ∞. Hence, for all x, 2 e3x √ +1 3x)2 = e · e( = √ ( 3x)2n e n! n=0 = e ∞ ∞ 21.We have 3nx2n . n! n=0 1 1 1 = , = x 2 + 3x 2(1 − u) 2(1 − (− 32 )) x where u = − 32 . So ∞ ∞ 1 3x 1 1 − un = = 2 + 3x 2 n=0 2 n=0 2 The series converges if |u| < 1; that is |x| < 2 . 3 ∞ n = (−1)n n=0 3n n x. 2n+1 ...
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