(a) Compute and simplify S(x) (the Fourier series of f). (b) Simplify the expression of S(7). (It will involve a sum). (c) Using Dirichlet's theorem, find S(x). (Hint: think about the periodic extension of f, what is f(n¯) and f(a+)). (d) Use (b) and (c) to show that 9. 16 25 WI

(a) Compute and simplify S(x) (the Fourier series of f). (b) Simplify the expression of S(7). (It will involve a sum). (c) Using Dirichlet's theorem, find S(x). (Hint: think about the periodic extension of f, what is f(n¯) and f(a+)). (d) Use (b) and (c) to show that 9. 16 25 WI

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter1: Fundamentals

Section1.5: Permutations And Inverses

Problem 4E: 4. Let , where is nonempty. Prove that a has left inverse if and only if for every subset of .

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Question

Let f be defined on (–x,7) by
0,
f(x) =
-T < x < 0,
0 <x< T.

(a) Compute and simplify S(x) (the Fourier series of f).
(b) Simplify the expression of S(7). (It will involve a sum).
(c) Using Dirichlet's theorem, find S(a). (Hint: think about the periodic extension of f, what
is f(T) and f(T*)).
(d) Use (b) and (c) to show that
1
1+
4
1
+
+...
25
-
6
n²
n=1
9.
16
IM:

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