Chem Differential Eq HW Solutions Fall 2011 27

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

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
Section 2.9 Uniform Convergence and Fourier Series 27 Solutions to Exercises 2.9 1. | f n ( x ) | = ± ± ± ± sin nx n ± ± ± ± 1 n 0as 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 .I f x ± = 0, then applying l’Hospital’s rule, we Fnd lim n →∞ | f n ( x ) | = | x | lim n →∞ n e - nx = | x | lim n →∞ 1 | x | e - n =0 .
Background image of page 1
This is the end of the preview. Sign up to access the rest of the document.

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 ....
View Full Document

This note was uploaded on 12/22/2011 for the course MAP 3305 taught by Professor Stuartchalk during the Fall '11 term at UNF.

Ask a homework question - tutors are online