Chem Differential Eq HW Solutions Fall 2011 27

# Chem Differential Eq HW Solutions Fall 2011 27 - f n 1 n =...

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Section 2.9 Uniform Convergence and Fourier Series 27 Solutions to Exercises 2.9 1. | f n ( x ) | = sin nx n 1 n 0 as n → ∞ . The sequence converges uniformly to 0 for all real x , because 1 n controls its size independently of x . 5. If x = 0 then f n (0) = 0 for all n . If x = 0, then applying l’Hospital’s rule, we find lim n →∞ | f n ( x ) | = | x | lim n →∞ n e - nx = | x | lim n →∞ 1 | x | e - n = 0 . The sequence does not converge uniformly on any interval that contains 0 because
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Unformatted text preview: f n ( 1 n ) = e-1 , which does not tend to 0. 9. ± ± cos kx k 2 ± ± ≤ 1 k 2 = M k for all x . Since ∑ M k < ∞ ( p-series with p > 1), the series converges uniformly for all x . 17. ± ± ± (-1) k | x | + k 2 ± ± ± ≤ 1 k 2 = M k for all x . Since ∑ M k < ∞ ( p-series with p > 1), the series converges uniformly for all x ....
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